120 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



but by (i) 



*-+!**-+. '".. 



hence 



Differentiating #/ by q s 



(13) J^L ^L-q^. 



Inserting these values in 



ar 



82 1 , , , 



+ y r 



,fa r \ ,d ftx r 



which, since 

 is equal to 



V I V ^ Xr V ty r 7 < ^ z r\ p 



* r \ r ^Qs r dq& r dqs) 



Hence we have proved by direct transformation the expression 



dt \dq 8 / dq s 

 to be equal to P 8 . 



