DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 

 Derivatives of the third order will be required. Write 



8V 



7o = 



* = 8/8^ ' 



8 = 



a** 



8\- 



a., = 



"= ^ ' 



so that 



|- /3 cZ?/ -f- t ofe 

 cZ/i = ft cZx + y dy + ft cZz 

 cZgr = j cZa 1 + /3j dy + ., 02; 

 rZZ> = 7o J.T + 8 dy + 7l <fe 

 cZ/ = /3j cZa; + 7l cZ# + y8 3 1?2 

 cZc = a. 2 dx -\- yS 2 dy + a 3 cZs 



Now, from the equation F = 0, we have, on supposing a proper value of v 

 substituted, an identity, which thus admits of being differentiated and leading to 

 identities. Thus 



d 3F 8F SF 3F 3F 



cTT + 3^ " + 0A A) + 07 a ' + si- y + a/ 



or say 



X + A + H^ + G ai + B yo + F^8, 

 and also 



Y + AA, + H yo + G/S! + B8 + F 7l + Cft, = 



Z + Ai + H& + G 2 + B ri + V/3, + Ca 3 = 



+ a- a '-' = Ol 



C = 



Consider the forms taken by these equations when the new variables arc used. 

 We have 



dc 



= a 



