174 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



The work done by the force X is 



[ T Xudt, 

 which lies between 



Uil Xdi and u \ Xdt, 

 Jo Ji 



where #, and u 2 are the greatest and least values of u during the 

 time r, i.e. it lies between u^ Af and 2 Af. If we now introduce the 

 supposition that Af, A?/, Af, AX, A//,, Ai/ are infinitely small, ^ and 

 u 2 are each equal to u, and the work done is wAf. In the same 

 way we may calculate the work done by the remaining forces 

 and couples. The total result must be equal to the increment of 

 the kinetic energy, whence 



aw op c? 



Now if the velocities be all altered in any given ratio, the 

 impulses will be altered in the same ratio. If then we take 



Aw _ Afl _ Aw _ Ap _ A&amp;lt;? _ Ar _ , 

 u v w p q r 

 it will follow that 



A| = A^_A? = AX = A/A = ^ = ^ 

 f V &quot;&quot; ? X &quot; /t z/ 



Substituting in (11), we find 



(2), 



since T is a homogeneous quadratic function. Now performing 

 the arbitrary variation A on the first and last members of (2), and 

 omitting terms which cancel by (1), we find 



f Aw + y&v + f Aw + XAp + /xAg + Mr = AT, 



