12 THE EQUATIONS OF MOTION. [CHAP. I. 



These equations might also have been deduced from (2), by multi 

 plying the latter by dt, integrating between the limits and T, 



putting X = I Xdt, &c., iff = I pdt, and then making T vanish. 



Jo Jo 



In a gas an infinite pressure would involve an infinite density ; 

 whereas no change of density .can occur during the infinitely short 

 time T of the impulse. Hence, in applying (19) to the case of a 

 gas we must put w = 0, whence 



/ Ttr/ / ~yl f y/ ff)(\\ 



In a liquid, on the other hand, an instantaneous change of 

 motion can be produced by the action of impulsive pressures only, 

 even when no impulsive forces act bodily on the mass. In this 

 case we have X , Y , Z each = 0, so that 



u u j- 

 p ax 



, 1 d-sr 



V V = -- -y- 



p d y 



1 (far 

 W W = --- 7- 



p dz 



.(21). 



If we differentiate these equations with respect to x, y, z, re 

 spectively, and add, and if we further suppose the density to be 

 uniform, we find by (9) that 



_._ 



&quot; 



The problem then, in any given case, is to determine a value 

 of -ar satisfying this equation and the proper boundary conditions* ; 

 the instantaneous change of motion is then given by (21). 



The Lagrangian Forms of the Equations. 



16. Let a, 6, c be the initial co-ordinates of any particle of 

 fluid, x, y, z its co-ordinates at time t. We here consider x, y, z as 

 functions of the independent variables a, 5, c, t ; their values in 

 terms of these quantities give the whole history of every particle 

 of the fluid. The velocities parallel to the axes of co-ordinates of 



* It will appear in Chapter in. that (save as to additive constants) there is only 

 one value of CT which does this. 



