442 MR. A. R. PORSYTH ON INVARIANTS, COVAR1ANTS, AND QUOTIENT- 



Hence three linearly independent solutions of (23) are u 2 u [ s u s u l 2 , u s u\ i' 3 , and 

 Wjtt'z u. 2 u\, say v lt v 2 , v 3 respectively. This proves the first part of the proposition ; 

 and for the converse we have 



= u i v \ u + v * u + v s u M i v i u i + v i u i + v a u s 

 = A.u l , 



so that, if the solution of (23) be known, then that of (22) can be derived. 



It is evident from the method of formation of (23) that it is the " adjoint " of (22), 

 see 52 ; the fundamental invariant is the same for the two equations, the change of 

 sign not affecting the invariantive property. We thus have a verification of the 

 proposition (2) of 62. 



83. Second, one immediately integrable form of the equation (22) occurs when 

 6 = C2~ 8 , c being a constant, for the primitive is 



u = AjZ"" + A 2 2"* + A 8 2" 



where m lt i 2 , m a are the roots of the equation m (m 1) (ra 2) = c. Another 

 occurs when = cz~*, in which case the primitive is expressible in terms of BESSEL'S 

 functions.* 



84. A third case, mentioned by BRIOSCHI (1. c., 5), occurs when vanishes ; we 

 may then take 



M! = 1, % = 2, % = 2 2 , 

 so that 



and therefore 



?i _ y* 



y* y$ 



or y^ = I//, which is practically equivalent to a general quadratic relation 



(*Xyi y y*f = - 



Since vanishes, we have for the uncanonical equation 2P 8 = 3 dP^/dx ; and, there- 

 fore, three linearly independent integrals of 



dy 3 dP 3 



*LOMMEL, 'Mathcmat. Annalen,' vol. 2, pp. 624-635, but without any notice of the adjoint relation 

 between the equations of odd order considered. 



