•^7^ 



1 



RUPTURE OF THE VESSELS WHICH CONTAIN THEM. 121 



mass compared with the column of fluid within the pipe, or the body of fluid 

 without. 



The instance of explosive forces on which I shall found their theory will be 

 readily understood. Conceive an elastic lamina a, of inconsiderable mass, 



and small modulus e to be placed between two 

 masses m and m , both large compared to a, but of 

 Fv^.2. which 7n! infinitely exceeds m. Let m be attached 



^ to a spring h that is retained at a distance s' from the 



point of repose, very inconsiderable with regard to 

 the deflections of a, but which, from the amount of 

 the modulus e of this second elastic body, will occa- 

 sion, when the retaining power is removed, an immense pressure on m, a, and 

 m' . Further, conceive m' subject to a small constant force that urges it in 

 a contrary direction to this pressure. 



The magnitude of e's and m' will allow us to consider these as the only ele- 

 ments concerned in the generation of the first motions, and thus when h has 

 attained the position of repose, or that where it has no elastic power, the vis 

 viva generated will be e o^; and as the motion of the mass m , attached to 5, is 

 now retarded, m' and m will separate; and the latter, after being urged to a 

 distance which, on account of the minuteness of the resisting force, is consi- 

 derable, will return to impinge on a, with the vis viva e a^; and if b has been 

 removed in the interval, this force must be destroyed by the sole resistance of a. 

 But the vis viva generated by a in traversing a distance 5' is equal to e' a'^; from 

 which it follows that when the returning strain occasioned by m' has been 

 wholly destroyed, we have 



eo^ = e' rj 



or 



^"V? 



as asserted in the sixth proposition. The use of the mass m' in this investiga- 

 tion will be readily seen by observing that it makes the velocity of the motion, 

 when h acts against a alone, quite independent of the mass of the latter; so 

 that a will stop when h has expanded through the extent of a vibration, and 

 the deflection thus attained will be ; or, in other words, the case assumed is 

 that of a heavy and powerful spring, acting against a light and weak one. 



VII. — 2 F 



