NOTE C. ART. 112. 



Let an infinitely great force X act for an infinitely short time T on 

 the solid in the direction of a;, so as to change the x-component of the 

 impulse from to + 8. The total work done by this force is 



T 



Xudt. 



J o 



If u lt u 2 be the greatest and least values of u during the time T, 

 this lies between 



u I Xdt, and u. I Xdt. 



Jo &quot; Jo 



i.e. between ujbg, and w 2 8 If 8 be infinitely small, u lt u z are each 

 sensibly equal to u, and the work done is uS. In the same way we 

 may calculate the work necessary to increase the remaining components 

 of the impulse by the infinitely small amounts 817, 8, 8X, &amp;lt;tc. Since the 

 work done is equal to the additional kinetic energy generated, we 

 have 



dT - dT ff 



= -r- ow + ... 4- -j op + .................. ( 1 ). 



du dp j 



Now if the motion of the solid be altered in any ratio, the impulse 

 will be altered in the same ratio. Let us then take 



and therefore 



