LXIV. 1867, pp. 1192-5. This is an extract from a memoir afterwards published in the Journal de Liouville: see our Arts. 203-20. Other parts of the same memoir are extracted in Comptes rendus, T. LXIII. 1866, pp. 1108-1111, and T. LXIV. 1867, pp. 1009-1013.

[203.] Sur le choc longitudinal de deux barres élastiques de grosseurs et de matières semblables ou différentes, et sur la proportion de leur force vive qui est perdue pour la translation ultérieure; ...Et généralement sur le mouvement longitudinal d'un système de deux ou plusieurs prismes élastiques: Journal de Liouville, T. XII. 1867, pp. 237-376, (the last two pages containing errata).

This is a long and theoretically very interesting memoir on the longitudinal impact of rods. It is the first complete treatment of the subject published. German writers have made some claim in this respect for Franz Neumann, who in his Königsberg lectures of 1857-8 dealt with the problem in somewhat the same fashion. But Neumann's investigations as first published in the Vorlesungen über die Theorie der Elasticität, 1885, pp. 340-346, are very insufficient and incomplete as compared with Saint-Venant's. Experimental investigations have been made by Boltzmann, W. Voigt, Hausmaninger and Hamburger with a view to testing the theory. Their results are not in full accordance with SaintVenant's formulae. I shall refer to certain points of difference in discussing the present memoir, but the articles devoted to their memoirs must be consulted for fuller details.

[204.] The memoir is divided into two parts, the first treats of the impact of two rods of the same material and of equal crosssection. It is divided into seven articles. The first of these (pp. 237-244) deals with the history of the problem. At the invitation of Coriolis in 1827 Cauchy had investigated the influence of the vibrations produced by impact in altering the translational energy of two rods; Coriolis having recognised that these vibrations. must be a source of loss in visible energy. Cauchy accordingly presented on February 19, 1827, a short note to the Academy, which was printed in the Bulletin...de la Société Philomathique, December 1826, pp. 180-182, and afterwards in the Mémoires de l'Institut. Cauchy treated only of the longitudinal impact of two

rods of the same material and section. He concluded that the impulse terminated whenever the two bars had not the same speed at their impellent terminals. This, as we shall see, is not true, and the conclusion vitiated some of Cauchy's results, the analysis of which does not appear to have been published.

Poisson in the second edition of the Traité de Mécanique (1833, Vol. II. pp. 331—47) also attacked the problem supposing his rods of the same material and cross-section. He used a double condition for separation, namely, not only that the bar which precedes shall have a greater speed at the impelled terminal than that which follows, but that the squeeze in both at the impellent terminals shall be simultaneously zero. This condition led Poisson to the singular conclusion that two unequal bars would never separate. He had forgotten that physically they can never sustain a stretch at the impellent terminals. In fact Cauchy's condition of excess of speed in the preceding bar is insufficient, and Poisson's additional one of no squeeze is superabundant. The true condition is clearly excess of speed at a time when there is zero squeeze at the impellent terminals, which can never sustain a stretch. It will also be necessary to shew that the bars thus separated are separated for good, and do not, owing to their vibrations, come again into

contact.

[205.] Saint-Venant's method of treatment is to investigate the vibrations of a bar, of which the initial condition is given by zero stretch throughout, and by speeds constant for each of the several parts into which the rod may be supposed divided. The first instant at which a zero stretch at the section between any two of these parts is accompanied by an excess speed in the terminal of the preceding section marks a disunion if the parts are not those of a continuous rod. In this manner Saint-Venant shews that if two bars of the same section and material are in impact the shorter takes ultimately and uniformly, while losing all strain, the initial speed of the longer.

This result was stated by Cauchy in 1826. Saint-Venant refers to the elementary proof of it given by Thomson and Tait in SS 302-304 of their Treatise on Natural Philosophy which in 1867 was in the press. His notice had been drawn to this proof by an article in The Engineer (February 15, 1867) due to Rankine

who, reviewing the extract in the Comptes rendus of Saint-Venant's memoir, had also given an elementary proof of one of his results for rods of different materials and cross-sections.

[206.] The second paragraph of the memoir (pp. 244-251) gives the general solution in finite terms of the equation for the longitudinal vibrations of a rod, when the initial speed and stretch of each point are given. The third paragraph deals with the special case of this when a rod of length a = a1+α2+a+... has these parts initially subjected to uniform speeds V1, V, V,... and uniform squeezes J1, J, J... etc. respectively (pp. 252–259). On pp. 254 and 258 we have diagrams which exhibit graphically in the special cases of two or three parts the speed and squeeze at each point of the rod during the motion. These diagrams are extremely instructive, and a similar method might be used with advantage in other cases of vibratory motions solved by arbitrary functions.

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[207.] The fourth paragraph is entitled: Problème du choc longitudinal de deux barres de longueur a, a, parfaitement élastiques, de même matière et de même section, animées primitivement de vitesses uniformes V1, V, sans compression initiale (pp. 259— 262). This applies the results of the preceding paragraph to the simple case of impulse above stated, taking V1> V, and a ̧ < α, Diagrams are given for the values of the speed and squeeze up to the time t given by kt = 2a, +2a, for the two cases 2a, <a, and a1<a,<2a. Here k velocity of sound (Ep). I have reproduced these diagrams reduced in scale on p. 140. Along the horizontal axis the values of kt are laid down, and along the vertical we have the various points of the combined rods, OA ̧ = α ̧‚Â ̧Â = α„ A ̧A In each area is placed the value of the speed and squeeze for that area, so that by means of the coordinates kt and x we can find the speed and squeeze of any point of the rod at any time. We see from this that at time t=2a,/k the contiguous terminals will be moving with unequal velocities V, and 1⁄2 (V1 + V2) but that this is only for the instant, and as there is no stretch at those terminals, the bars will not separate. They afterwards move till t = 2a/k with the same velocity at the contiguous terminals and no squeeze. The impulse is terminated, but the bars do not yet separate.

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Unequal speeds occur again when t=2a/k, and now the upper bar has a negative squeeze, j = (V,- V1)/2k at the

impelled terminal. Hence the solution no longer holds, and we have to treat each bar from this epoch as a distinct one. The bar a moves obviously without strain and with the speed V2 which the bar a, initially had. To deal with the bar a,, we have to distinguish two cases. Let us suppose:

1

(1) 2a,<a2 We have to enquire how a bar of which a

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portion 2a, has initially a speed v = {(V1+ V ̧) and a negative squeeze j= (V, − V1)/k, and a portion a,- 2a, a speed v = V2 and a squeeze j = 0 subsequently moves. This has been ascertained in the second paragraph of the memoir and is represented by Saint-Venant in the accompanying diagram.

1

with

We see at once that after t = 2a,/k the terminal moves with speed V, and therefore separates from the terminal of a1 speed V-V. This lasts till t = 2 (a,+ a)/k, when what happened at time t = 2a,/k repeats itself and the terminal moves with speed V,, i.e. with the same speed as the terminal of a,. Thus it alternately moves with greater and equal speed, or the two terminals never again come into contact.

1

(2) α1<a2<2α, We have to enquire how a bar of which a portion 2a,- 2a, has initially a speed v = (V1 + V2) and negative squeeze j=-} (V1- V2)/k, and a portion 2a, -a,, a speed v V, and squeeze j=0 subsequently moves.

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The motion is represented in the first diagram on p. 142, and we see that after the time t = 2a,/k these bars never again come into contact.

[208.] The second diagram on p. 142 represents the whole motion of the two bars supposing them to be endowed with a uniform velocity perpendicular to their lengths during and subsequent to the impact. The full lines give the paths of various