DKKIVATJVES ASSOCIATED WJTH LINEAR DIFFERENTIAL EQUATIONS. 441 



/''/nation of the Third On/' r. 



81. The general results obtained in 30 show that, by the solution of 



K*}=*P* 

 the equation 



is transformed to 



where 

 and 



= 0, 



2 "e=P 8 -i^ : 



and, if we write z = 0~ 2 , the equation determining is 



(22) 



The form (22) is the canonical form of the cubic. 



82. First, if the solution of (22) be known, then that of 



cPv 

 dz* 



(23) 



can be derived from it, and conversely. For let M,, ?tj, u 3 be three special and linearly 

 independent solutions of (22); then we have 



= A, 



where A is a determinate constant. Introducing a new quantity v 3 , defined by the 

 equation 



V S = MjW ! 2 



we have 



= 6 



MIXV lAXXVIII. A. 



3 L 



