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equal either 10 or 11. If then v÷n, and k be the last odd number personal equation common to the whole race is equivalent to making which does not equal n, we have

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(For the algebraical solution of binomial equations, see the works of Gauss or Legendre on the theory of numbers, or Murphy's Theory of Equations in the Library of Useful Knowledge.)

EQUATION OF PAYMENTS, an arithmetical rule, for the purpose of ascertaining at what time it is equitable that a person should make payment of a whole debt which is due in different parts payable at different times. This rule is now of no practical use, as it rarely, if ever, happens that it is considered necessary to equate payments. Sums of money due at future periods are generally secured by bills of exchange or by promissory notes, and when the date of payment is altered, it is usually immediate payment which is contemplated. [DISCOUNT.]

EQUATION, PERSONAL. It is a fact which has now been for some years established, and which might reasonably have been suspected, that different persons, attempting to observe the precise moment of a phenomenon, by means of a clock which beats seconds, do not agree exactly in their results, but differ generally in one and the same way, one of the observers being almost always a little before the other in the moment which he assigns to the phenomenon. If this had not been the case, if one of the observers had been about as often before the other as behind him, the difference could only have been considered a simple casualty. But, looking upon observers, it becomes obvious that the cause is in the organs of the men themselves; and that physical constitution, temperament, habit, &c., make differences between one person and another.

the constant occurrence of a difference of one kind between two

and 0.4.

all the clocks wrong by the same quantity. Suppose, for example, that every person suddenly received an addition of one second to his personal equation, or began to note phenomena a second later. The astronomers would then begin to find the clocks a second too fast, in comparing observation with prediction; as soon as the clocks had been rectified every thing would be as before.

The first notice we have of personal equation is in an announcement by Maskelyne, in the volume of Greenwich Observations for 1795. He tells us that he was obliged to part with one of his assistants, because the latter, who had till then always agreed with him in his observations, suddenly began, in August, 1794, to observe half a second later and that in January, 1796, the difference amounted to 8-tenths of a second. Maskelyne inferred that his assistant had contracted some bad habit of observation: it is now very well known that age causes persons to observe later than they did before, though it is not usual for the habit to undergo such sudden changes as in the above case. In 1823 Mr. Bessel, at Königsberg, ascertained that he was in the habit of observing phenomena as much as 1.22 before his assistant Mr. Argelander. The latter left his post to take charge of the observatory at Bonn: but, passing through Königsberg in 1832, Mr. Bessel took the opportunity to make some further comparisons with him; and it was found that the above quantity was reduced to 1:06. Age had brought them nearer together. A close trial of the subject was made by MM. Quetelet and Sheepshanks, 1838-1841, in determining the longitude of Brussels by transmission of chronometers between that place and Greenwich. This method of course requires the most careful transit observations at both places, and the personal equation* becomes of considerable importance. It was necessary that some observer should compare himself with M. Quetelet at Brussels, and with the assistants at Greenwich. This was undertaken by Mr. Sheepshanks, and the result ascertained was, that he came, one observation with another, 45-hundredths of a second behind M. Quetelet, and 27, 35, and 24-hundredths before MM. Main, Henry, and Ellis, severally. The result was that the longitude of Brussels was found to be threequarters of a second less than it would have been supposed to be if the difference of personal equations had been unknown. (See a memoir on the difference of the longitudes of Brussels and Greenwich, by MM. Quetelet and Sheepshanks, Mém. de l'Acad. Roy. de Bruxelles, vol. xvi.)

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EQUATIONS, DIFFERENTIAL, and EQUATIONS of DIFFERENCES. The difficulty in this case is the inversion of the processes of the Differential Calculus and the Calculus of Differences. day dy We give an example of each case :-x, is a differential dx d.c equation. The question asked is, what is y, that function of x, of which it is the property that the first differential coefficient subtracted from the second will always leave x.

Ayy+1, is an equation of differences. The question asked is, what must y (understood to be a function of a) be, in order that an increase of a unit in the value of shall increase y by y + 1. This is in reality a simple functional equation, as follows. Required .c

so that

• (x+1)− p (x) = x + 1.

The

The two classes of equations, thus briefly noticed, include in their history that of most of the mathematico-physical sciences. progress of the theory of gravitation since Newton is contained in successive attempts to solve certain differential equations. All ques tions of dynamics, electricity, the theory of light and heat, &c. &c., resolve themselves at last into the solution of differential equati Works on the differential calculus contain but little on this subject, its utility considered; and it is to the applications themselves that the student must look for further information.

We shall now give a slight synopsis of results, such as may be useful to the advanced student, as a guide to his reading, or for reference. We have not room either to teach the subject or to illustrate it by examples. We may refer to the following works:-Moigno, 'Leçons de Calcul Differentiel,' &c., vol. ii. Paris, 1844, 8vo.; De Morgan, Differential and Integral Calculus,' London, 1842, 8vo. (Lib. of Useful Knowledge); various numbers of the Cambridge Mathematical Journal,' Cambridge, 1839, &c. 8vo.; Brooke, Synopsis of the.. Formula.. of Pure Mathematics,' Cambridge, 1829, 8vo.; Peacock, Herschel, and Babbage, Collection of Examples,' &c., Cambridge, 1820, 8vo.; Gregory, Examples of the Differential Calculus,' Cambridge, 1841, 8vo.; Boole, on 'Differential Equations,' Cambridge, 1859. We must confine ourall reasonable limits. selves to a selection from striking points, or our article would exceed

Personal Equation is a name given to the quantity of time by which a person is in the habit of noting a phenomenon wrongly; and it may be called positive or negative, according as he notes it after or before it really takes place. Thus if A and B are severally in the habit of noting events 3-tenths of a second after and 4-tenths of a second before they take place, their personal equations may be described as being +03 The absolute personal equation of any one is a thing undiscoverable; since we can only refer one human observer to another, and note the difference of their times of observing the same phenomenon. If we could cause a thousand persons to note a given definite phenomenon by one clock, and if we could take the mean of all their results, we might say that there is very strong reason to presume that the mean is the time which perfect organs would have noted: for we may think that the chances are much in favour of human imperfection being, in the mass, as much of one kind as of the opposite. But a little considera-(x, tion will show that this reasoning is not to be relied on. It may be that the whole race has, by its constitution, a rather large personal equation of one or the other kind: for we can only see the differences, without knowing upon what quantities they are differences. This question is, however, practically immaterial; for any given amount of

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1. Let there be a function p (x, y, a, b, c .), or p, containing the variables x and y and the n constants, a, b, c,... If we make = 0, we tacitly require that y should be a function of x. If between y...) = 0, and the result of complete differentiation, o' (x. y, ...) =0, we eliminate one of the constants, we get a new equation contain

It is usual to call the difference between two observers the personal equation; but this ought to be called the difference of their personal equations, or their personal difference. The personal equation of an observer is, properly speaking, the difference between him and the average of the human race.

ing x, y, y (or dy: dx) and all the rest of the constants; and according as we eliminate one or another constant, we have one or another of as many such equations as there were constants. These are called differential equations of p=0, and they are said to be of the first order. 2. The word order refers to the number of differentiations, the word degree to the highest power of the highest differential coefficient which enters. Thus y++y=x is a differential equation of the second order and the fourth degree. Let the accents always denote complete differentiation with respect to x. For partial differentiation we shall | do аф save room by writing as follows:--p' for p' for dx y dy'

&c.'

3. If we eliminate two of the constants between p=0, p′ = 0, "=0, we have a differential equation of the second order, and so on; there are altogether in (n-1) differential equations of the second order, if we have a constants. And generally, when there are n constants, there are as many differential equations of the mth degree as there are ways of taking m out of n things. Thus there is only one equation of the nth degree.

4. If there be a system of p equations between p + 1 variables, so that p of the variables are functions of the remaining one, and if we differentiate each equation once, we can eliminate p constants, and form p equations of the first order. With p second differentiations, we can eliminate p other constants, and so on. And if there were p+q variables, we should have similar systems with q independent variables.

or

5. Equations of differences are formed in a corresponding way by taking differences instead of differential coefficients. Thus suppose u, means u, a function of x, and (x, uz, a)=0. Change x into x Ax, usually into x + 1, and let u, then become ux+19 ux +Aux. If we eliminate a between p (x, U z, a) = 0 and p (x+1, ux+1, a) =0, we have an equation of the form ¥ (x, u x, U x + 1) 0, or (x, u ̧, ▲ux) = 0, as we please, and this is an equation of differences of the first order. But exactly the same equation would be obtained, if for a we had substituted any function of cos (2x), oг of cos (2π÷Ax). On this point see INVARIABLE.

=

6. When there are more than two variables, say three, it is possible to form an equation in which an arbitrary function of a definite function of two of them shall be eliminated. Thus, if {x, y, z, ya Φ (x, y)}=0, in which z is an implied function of x and y and a (x, y) a given function; we differentiate with respect to x and y separately, and produce three equations involving x, y, z, zx, y, a (x, y) and a (x, y). From these three eliminate the two last, and ', and we get an equation involving only x, y, z, zx, y, in definite forms. This is called a partial differential equation.

7. The analogies between the constants of a common differential equation and the arbitrary functions of a partial one must not be relied on as capable of being carried all lengths. It is not, for instance, universally true that two arbitrary functions may be completely eliminated by differentiations of the second order.

We now consider common differential equations of the first order. 8. Let y= (x, c), and y'=p (r, c), and let elimination give y' = x(x, y), the differential equation. There is another mode of arriving at the same result. Let y= (x, c) give c= (x, y), or 0=' + ', . y. The constant here disappears by mere differentiation, and x (x, y) must be identical with — P', :'y.

y.

9. The difficulty of returning to the primitive from the equation y' = x (x, y) consists in that of reducing it to the form ♣', + Þ'y. y' = 0. Generally speaking, common factors are made to disappear, and the restoration of these factors is a problem of exactly the same difficulty the solution of the equation.

10. The quantity P+Qy, P and Q being functions of x and y, is integrable at once when P, Q. When this condition, which is usually called the criterion of integrability, is satisfied, the integral is

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11. The equation y=x (x, y) can have no other primitive with an arbitrary constant, except y=p (c, c) from which it was derived. But it may have another solution which has not an arbitrary constant, or even more than one: these are called singular solutions. A singular solution makes x', and x', infinite if y be substituted from it in terms of x. It also makes ', and ',. infinite in the same manner. All the solutions of either or both p = 0 and ', 0 are singular solutions; and all the singular solutions are thus found. But though the most prominent and useful singular solutions are sure to be obtained from x' = ∞ and x'y =, these equations may not contain all singular solutions, nor do they give nothing but singular solu

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tions. But any solution of xy which makes X'a + X'y x finite is a singular solution of the equation y'

= x.

12. The singular solution may also be found from the primitive y= (x, c), as follows. Eliminate c between y = (r, c) and 0='e (x, c), the result is the singular solution. This rule and the last are subject to exception, when the singular solution takes the form x = constant or y = constant.

13. The geometrical character of a singular solution is as follows. If y = (x, c) be the equation of a family of curves, that is, of one curve for each value of c, the differential equation y=x (x, y) also belongs to every member of that family. The singular solution is the equation of the curve which touches every one of the family of curves.

14. The equation y=yx + (y) is historically remarkable as having led to the theory of singular solutions: it is called Clairaut's form. The complete primitive is y=cx+oc, belonging to a family of straight lines: the singular solution is found by eliminating c between this and x+¢'c=0.

15. Singular solutions are either intraneous, contained in the general solution as particular cases; or extraneous, not so contained. These two species do not differ in geometrical character. A remarkable theorem of Cauchy discriminates between them, as follows:Let y=P be a solution of y'=x(x, y), and let ẞ be any small quantity. If, then, a being constant,

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be finite, y=P is an extraneous solution; if infinite, intraneous. 16. It is not often that the factor which makes an equation integrable can be recovered in any general form. As soon as the solution is obtained, it can always be found; but the only use of it is to find the solution. There is always an infinite number of such factors, any one of which will do. When in P+Qy', the criterion P',-q', is not = 0; yet if Q-1(P',-Q') be a function of x only, and not of y, and if Q-1 (P'y — Q's) dx=V, then eV (P+Qy') is integrable,

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to which common integration can be applied. The equation is now reduced to one in which the variables are said to be separated. 19. It is hardly necessary to say that the variables can be imme. diately separated in P+Qy=0, whenever P and Q are both of the form ¢x ×¥y.

20. It often happens that, when direct processes would require the previous solution of an equation of indefinite form, indirect processes will succeed in reducing the question of solution to one of elimination. Thus x=f (y') is solved when y' is eliminated between this and

y=yfy' fy' dy'+c,

and y=fy' is solved when y' is eliminated between it and

x='$(-)-S1 (-) d'+e

where x' means 1: y'.

21. The equation y=x oy′+4y' can be reduced to elimination, thus: Let y'z, then

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which is of the form integrated in § 17. Eliminate z between the result of this and y=xoz+ 42. 22. The equation y= (x, y') can be reduced to a form of the first degree and elimination. Let y'z, then z=P+Qz, where P and Q are p' (x, 2) and o'z (x, z). If this equation can be integrated, we eliminate z between the result and y = (x, z). The form x=(y, y') may be treated in a similar manner. Or generally, let the equation be (x, y, y')=0; that is, (x, y, z) = 0, which gives by differentiation a form P+Qz+ Rz' = 0. If from this we eliminate y, by substitution from the preceding, we have an equation of the first degree as to ; from which, if integration be possible, elimination will determine the relation between x and y.

23. Any differential equation whatever may have a chance of reduc

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24. Various substitutions will sometimes aid in the solution of an equation, as y=ux, y=u ̄1, =u", y=e", and so on: sometimes, when couple of expressions of the form ax + by +c, a'x+by+c', occur, it will be convenient to introduce two new variables derived from equating the preceding to Au + Bw and A'u + B'w. We have given the most effective general methods; the number of transformations * which have succeeded in particular cases is very large.

25. When the equation is of the first order, but of a higher degree, as Pay'" + P-1 y'n−1 + + P1y′+ P= 0,

...

the theoretical mode of proceeding, occasionally convenient enough in
practice, is to find the n values of y', say y'= (x, y), y' =Q2 (x, y),
&c. If these can be separately integrated, giving, say (x, y, c)
¥2(x, y, c) = 0, &c., then the complete primitive of the equation is
41 (x, y, c) × 42 (x, y, c) × .... = 0.

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in which A,, &c., are of the form (y: x). Let y = tx and find, in terms of t, the several values of y' or txt', from the equation. Now t+xt' = pt, is an equation in which the variables can be separated. Solve each equation, and proceed as in the last.

27. The only equation of differences of the first order which can be generally treated, at present, is the form Au-P, Q, where P, and Q, are functions of a, or its corresponding form Uz + 1 − P z U z = Q . Supposing x to be an integer, which is what is usually required, let Zu, denote um + Um + 1 + ... + u-1, where m is an arbitrary integer chosen to start from. Accordingly Au The solution of Ux+1-Px Ux Qz is

=

Ux = PmPm+1 • • · · Pz− 1 °

= Ux•

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Σ

Qx

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and (m+1)/xTM P ̧dx=x+1 Pn −Pm+n+1

If the integrations in the last section be made by this last rule as they arise, the results will be given in terms of Po, P,, &c., and there will be no difficulty about the constants, which in the ordinary mode may appear to enter in too great numbers.

first which can be integrated in general terms. The equation y′′=qy gives

32. There are very few cases of equations of higher orders than the

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y'2=c+2 Spydy

Again, y'+Qy'R, Q and R being functions of

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in which P, &c. are functions of x only. If a solution be found which
solution, but is called a particular solution. It is a most important
has not n distinct arbitrary constants, it is not the most complete

property of the linear equation that if n distinct particular solutions
can be found, say y=Y1, y=Y, &c., then the complete solution is
y=C1Y2+C2Y +....
..+CAYA

where C1, C,, &c., are any arbitrary constants.

34. The most remarkable case is that in which the coefficients are + any = 0. If the equation constants, as in y(") + ay (n−1) + . . ., solution is k" + a ̧k2¬1+...+an =0 have n unequal roots, a, 8, 7, &c., the general

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28. Equations of differences in their most general form have solu- A pair of imaginary roots, μ± √-1, contributes to the solution tions which appear to resemble the singular solutions of differential equations but there are important points in which the resemblance fails. (Differential Calculus-Library of Useful Knowledge,' p. 738.) 29. Let y() stand for the nth differential coefficient of y. The following expression

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+ (P2 - 2 — P′n−1 + P′′n ) y(n−3) + One remarkable case is this:-each term Py(m) is absolutely integrable when Pm is a rational and integral function of x of a degree lower than the mth. Thus yiv, xyiv, ayi, x3yiv, are all integrable functions; and ayiv, xyiv, are also integrable.

30. The equation y(a) px is not only completely integrable, but it may easily have all its intermediate differential equations found. For instance, suppose ype. Multiply this successively by 1, x, x2, x3, x+, and, by the last section, we have five integrable equations, leading to the five differential equations of the fourth order which belong to y = px. Treat the first result in the same manner with 1, x, x2, x, the second with 1, x, x2, the third with 1, c, and the fourth with 1, and we have the 10 equations of the third order. Proceed in the same way with the results, taking care never to let any multiplication enter which raises the coefficient as high as the order of differentiation which it accompanies, and then will appear successively 10 equations of the second order, 5 of the first, and finally the original primitive.

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No one differential equation is of so much importance as this.
36. When the equation is linear in all but an independent term, as
y(n) +P ̧y(n−1) +....
..+Pay=X

x being a function of x, its solution can be deduced from that of the
linear form, in which x=0, by a method the importance of which will
justify some account of it in a separate article. [VARIATION of PARA-
METERS.] We give the following result of it: y" + a2y=x gives
a y=sin axx cos axdx—cos ax fx sin axdx

+ K cos ax+L sin ax.

37. The completed linear form (take the second order for an instance) Poy" + P1y +P2y=x can be integrated (§ 29) one step if both sides be multiplied by the factor M, which satisfies the purely linear equation MP-(MP)+(MP ̧)" = 0

This is another differential equation of the same order, but the difficulty is much reduced by it. The general solution of the last is not wanted; any particular solution will do. That is, if a particular solution of every linear equation could be found, the general solution of every completed linear equation could be deduced. And it will generally happen that when one particular solution can be found, to make an integrating factor, enough of them can be found to give the complete solution of the original equation.

The student must not be discouraged by finding that he does not succced in solving cases which require detached artifices in the same manner as the 38. The form (y, y', y")=0, in which a is wanting, may be reduced writer of the elementary work before him. The art of solving equations is made to an equation of the first order, by making y'=z, when it becomes evident the art of constructing equations which can be solved is behind the curtain. There are few or no mathematical works, with many examples, to the The word particular is here opposed to general. Note this, because some writers of which the proverb may not be applied, that "those who hide know writers have used particular solution to signify what is now generally called

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(y, z, zdz: dy). If z or y' be thence found in terms of y, the variables can be immediately separated: if y be found in terms of 2, we eliminate z between y=4z as found and x= (4'zdz : 2).

49. When the forms are

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39. As an à priori mode of constructing linear equations and their 41 &c. being constants (a case which is often useful) determine the solutions, note the following: If p (x, y, y', ....y()) have for its solution y= (x, a, b, c, . . .), a, b, c, &c., being arbitrary constants, and if Po, P1, P2, &c., be the differential coefficients of 4 with respect to and the two values of z from y, y', y", &c., considered as independent variables (with substituted in them instead of y: then the solution of the linear equation P ̧u+Р ̧u' + Þ2u" +....Pnu(n) = 0

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42. In this way the solution of y'=px. y is connected with that of +22=px. When pax, it is called Riccati's equation, and it can be integrated in finite algebraical terms whenever m=-4k : (2k± 1), k being a positive integer.

43. If only a particular solution of y" + ry+Qy=0 can be found, then the general solution of y" + Py' +Qy+R=0 can be found from it. Let y=y be the particular solution of the first, and let log. W= rdx: the general solution of the second is

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46. Any number of simultaneous equations which leave only one independent variable can be reduced to equations between two variables. For instance, let there be three equations between x, y, z, t, and differential coefficients of x, y, z, with respect to t. Now, looking upon x, x', &c. y, y', &c. (excluding z, z', &c.), as so many independent quantities, differentiate the three equations with respect to t.

There

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and find x and y from x+0,y=z1 , x+02y=220

50. Also, when the simultaneous equations are linear, with constant coefficients, proceed as follows. Say there are four variables, x, y, z, t; and three equations. Assume x=eat, y=bet, z=ceat, and substitute: the result will be, after division by eat, three equations of the form (a, b, c)=0, in which b and c only enter in the first degree. Honce the values of a can be found by an equation whose degree is the sum of the orders of the original equations, with one value of b and c for each value of a. Let the values be a, a,, b1, b, c, c2, &c., then the complete solution is

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c, &c. being arbitrary constants.

....

51. Whenever the solution of a differential equation depends upon that of an ordinary equation, and supposing the roots to be all different, the form of the solution is

x=c1 (at) +∞2 $ (a,t) + . . . .

the solution, though still a solution, ceases to be the complete solution
when there are any sets of equal roots. If there be m roots equal to a,
and if p' x, p" x, &c., be the successive differential coefficients of px,
this set of roots contributes to the value of x the terms

K (at) + K ̧ o'(at).t+...
~ • • • • • • + K m −1 p(m−1) (at) . tm−1

of which a particular case has been seen in § 34.

52. We now look at one equation with three variables, two of which are therefore independent. Let xdx+ Ydy + zdz=0 be the equation. This may have arisen direct from p(x, y, z) =c, without alteration or loss of factors, &c.: and this will be the case when x'y=Y'z, Y' z = Z'y, z'x',, and then only. To return to the primitive, integrate xdx + Ydy (§ 10) as if z were constant, and let the result be P. The integral of xdx + Ydy + zdz is then

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Or thus when the criteria are satisfied, the integral (omitting the usual commas, to save room) of

p (xyz) dx+X (xyz) dy +¥ (xyz) dz is

S* • (xyz) dx + s' x (ayz) dy + S. ♦ (abz) dz

a

in the first of which y and z are treated as constants, and z in the second.

53. If the criteria be not satisfied, a factor м may possibly exist, such that MX da + MY dy + Mz dz shall be integrable. But this cannot happen unless the following criterion be satisfied:

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e theni ntroduced two new differential coefficients of x and y, and three new equations: or the number of equations gains one upon the number of quantities. Repeat the process, and the same thing happens again, and so on. Consequently a step must arrive at which the number of equations is made one more than the number of quantities. All the quantities can then be eliminated, and there results one equation between z, z' &c., and t. The same thing may be done with x and y. The process will also reduce the number of equations to as many as there are independent variables, whatever that number may be. The order of the result is the sum of the orders of the original equations. 47. In the simple case of Ay' + Bz'+C=0, Py' + Q+R=0, when A, P, &c., are functions of x, y, z, and z' is dz: dx, &c., we may pro-point of one of these surfaces being taken, and any other point ceed by eliminating y' and forming say z'=M, a function of x, y, z. Hence M+M', y' + M', z', from which we may eliminate y and y' by the original equations, and thus produce an equation of the second order to determine z in terms of x. And the same for y in

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But if the criterion be not satisfied, then there is no equation what-
soever between x, y, z, which always produces xdx+xdy + zdz=0.
54. The equation xdx + Ydy + zdz = 0, when produced from
(x, y, z)=c, is the equation of a family of surfaces, the individuals of
which are defined by the different values of c.
That is to say, any

infinitely near to it, the equation is satisfied by x, dx, &c., as derived
from these points. But when the criterion is not satisfied, then
xdx + ydy + zdz still belongs to any surface in the following limited
sense. On any surface, and through any point of it, a curve may be
drawn such that the equation xdx + &c. is satisfied if the two con-
tiguous points first named be taken on that curve.

55. When there are three variables and two equations, with two constants, as in p (x, y, z, a, b) = 0, ¥ (x, y, z, a, b,)=0; the most ready theoretical mode of imagining the differential equations formed is the reduction to the form a= (x, y, z), b=¥ (x, y, z), and differentiation. Two equations of the form rdx+Qdy+Rdz=0 are thus produced; from which, by combination, may be derived an infinite number of pairs of the same kind, answering to the infinite variety of pairs which can be produced from the primitive equations. This corresponds with the obvious geometrical fact that one curve may be the intersection of an infinite number of different pairs of surfaces.

56. A partial differential equation, such as 4 (x, y, z, z′ x, z′ y) = 0, belongs to an infinite number of surfaces distinguished by the forms of the arbitrary function which enters into the solution. The most general method of proceeding is as follows, which supposes that a particular solution is found having at least two new arbitrary constants. Let f (x, y, z, a, b)=0 be such a solution, and call it a primary solution. Make b= Fa, where Fa is any function of a, at pleasure. Then the general solution is

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The arbitrary character of ra introduces an arbitrary function.

The geometrical meaning of this is as follows: The primary solution, a and b being independent, is the equation of an infinite number of families of surfaces: when b is made=Fa, the equation is restricted to one of these families; and each case of the general solution answering to one form of Fa, is the equation of the surface which touches every individual of the family throughout the extent of some curve. Every surface which comes under the general solution is then tangent to a whole family of primary solutions. But it must not be forgotten that every primary solution is also a case of the general solution, so that there is no primary solution but what is also tangent to a whole family of other primary solutions.

57. There is generally a surface which touches all the surfaces of the general solution, and is a singular solution of the partial differential equation, not contained in the general solution. It is found by substituting in ƒ (x, y, z, a, b)=0, values of a and b derived from

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can be integrated in the form a=P, b=q, c=R, e=s, described as before. The complete solution is the system

F,(P, Q, R, S) = 0, F,(P, Q, R, S) = 0, F2(P, Q, R, S) = 0, where F1, &c., are symbols of any functions whatever.

60. Nearly all that can be done in this part of the subject depends upon cases of three variables and their connection with surfaces. The following notation is in universal use: Having z, a function of x and y, let z and z', be denoted by p and q, and the second differential co-efficients z" xx, 2" * z" by r, s, and t.

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61. Let z=px + qy+f(p, q) be the equation. A primary solution is z=ax+by+f(a, b). The general solution, deduced as in § 56, gives the equation of all developable surfaces, if ƒ be arbitrary. The form q=p, being a given function, has z=ax+pa.y+b for a primary solution, and belongs to a particular class of developable surfaces. Both these equations, and all developable surfaces, satisfy the equa

tion rt-80.

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(ay', y'), giving, say y=4(x, b). Then z= =(y+ax, b) is a primary solution of the given equation. 65. Let (p, x) = 4 (q, y). Let (p, x)=a, 4(q, x)=a, give p=♣ (x, a) q=¥ (x, a). Then eliminate a from [ { 4(x, a) . dx +¥ (y, a), dy} +xa

0

2=

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If the resulting equation can be integrated into z=4(x, y), then the original equation is integrated by eliminating x and y from x=P y=Q z=PX+QY-Z

67. Linear equations of differences with constant coefficients, such as Uz + 3 + AUx + 2 + bụ x + 1 + cu2 = 0

may be solved in a manner corresponding to linear differential equations by assuming u2 = m* and obtaining its particular solutions by means of m3 + am2 + bm + c = 0.

We have not touched on the application of the calculus of operations, nor on that of definite integrals, subjects which are now of very great importance. On these points see OPERATION, INTEGRALS, DEFINITE. See also FUNCTION, ARBITRARY.

It must be noticed that a great portion of the most important part of the subject of differential equations cannot enter in any work on pure mathematics. The physical subjects of gravitation, heat, electricity, &c., depend so much on certain differential equations that, by common consent, the details are referred to works on these subjects, and do not appear separately.

EQUATIONS, FUNCTIONAL. In this case the question is to find the form of a function which will satisfy certain conditions. For instance (x) = x + 1. Here the question asked is, what is that algebraical expression which will be increased by 1, whatever may be the value of x, by changing a into x. [FUNCTIONS, CALCULUS OF.]

EQUATOR and ECLIPTIC, the two principal circles of the sphere. The first is that circle of the apparent celestial sphere which is in all points equally distant from both poles; the second, the circle through which the sun appears to move. The equator is so called from being the circle on the arrival of the sun at which the day and night become equal. The ecliptic derives its name from being the circle on which (or near which) the moon must be in the case of an eclipse, [SPHERE, DOCTRINE OF THE.]

EQUATOR, MAGNETIC. [MAGNETISM.]

EQUATORIAL INSTRUMENT. This name is generally given astronomical instruments which have their principal axis of rotation in the direction of the poles of the heavens. When the purpose of a machine of this nature is simply to carry a telescope, it has been called a machine para lactique or parallatique by the French, and sometimes polar axis by English writers; but we shall include both in this article.

The

The complicated system of circles which formed the astrolabe of Hipparchus, described by Ptolemy (Almagest.,' lib. v. cap. i.), was made moveable on two pins, which marked the places of the pole in a metallic meridian circle, and thus may be called in some sort an equatorial. There is an excellent plate of the astrolabe in the titlepage of Halma's translation, tom. i. This instrument and the copies which were made of it afterwards, according to Ptolemy's description, by the Arabs and by Walther of Nuremberg, were designed for observing the longitude and latitude of a heavenly body directly. torquetum of Regiomontanus was for the same purpose, but surfaces were used instead of axes to determine motions, but we know not whether it was ever actually made. Tycho seems first to have seen the immense superiority of the simpler instrument, which sufficed for determining right ascension and declination: and the genuine equatorial is therefore due to him. In his Astronomica Instaurate Mechanica,' Noriberge, 1602, we find the figures and descriptions of three equatorial armilla' of different sizes and constructions; in one, the diameter of the meridian circle was 7 cubits, or 10 feet. (For Tycho's equatorials see ASTROLABE.) In the Rosa Ursina' of Scheiner, Bracciani, 1626-30, p. 350 et seq., there is a plate and description of an equatorial 64. Let z= (P, ). Solve the common differential equation y mounting, invented by Gruenberger, to be used with a lens or telescope,

62. For the method of dealing generally with the equation of the second degree, Rr + S8+ Tt=v, see Differential Calculus,' in 'Library of Useful Knowledge,' p. 719. It would hardly yield a short account for

a work of reference. 63. Let rat. solution is

Of this very important equation the complete
z=4(y+ax)+4(y—ax).

and being any functions.

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