DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 



39 



These six identical relations reduce the twenty equations in the two foregoing 

 columns to fourteen independent equations ; hence the fifteen derivatives r can be 

 expressed in terms of one of them, say r 5 , and of derivatives of , . . , , S ; and the 

 value of r- a is dfi^du -r- dz/clu. These expressions for the other fourteen, in terms 

 of r B , are 



- ( p -. l (I 

 1 ( 1 \ dy dx 



, dft a 

 2+ dy 



7 / 



T 



r q - 

 5 P 



l_ 

 ?5 2> 



'-P 



'p dx 



10 



_ oj _ 



P 



dy 



13 



J- _U J- / ^' ^ -4- - ^ 



- r 5 - - 3 ( ^ '- ^ rf " ' ^ tlx 



15 



5 2? '/ dx p dx 



11 7 Q 1 ,1 Q 



1 Up, 1 rtp.) J 



T- 5 -f" "^ " "~7^ + 



20. When these values are substituted in the six equations, and terms are collected, 

 it appears that the terms in r 5 in the six equations are 



A A A A ? A , A 



respectively. For reasons which, being the same as before, need not be repeated 



