DERIVATIVES ASSOCIATED WITH LIKK.Yft DIFFERENTIAL EQUATIONS. 413 

 Now, in $ 39 we have seen that the Jacobian J of . and 8^, is given by 



a result enunciated for the case in which 8, (there 8 P ) is supposed a priminvariant, 

 though, in the proof, no such limitation was introduced. This may be applied to the 

 consideration of (a) by writing 8, = 8 P , , l ; the functions 8^ ., l and 8 Mt , t l are then 

 cuhinvariants (composite, moreover), and therefore J can be expressed in terms of 

 invariants of the first three classes. Thus from (a) no proper quartinvariants arise. 



The same formula may be applied to the consideration of (b) by writing 8, = 8 p , , ; 

 the functions 8 Aiir)] and 6^,i are again cubinvariants (composite, moreover, if X and p 

 differ from p) and so J can be expressed in terms of invariants of the first three 

 classes. Thus from (b) no proper quartinvariants arise. 



These two results can also be deduced as follows. For (a) we take 



j = (x + M + 1) B A , M . , e;,,, , - (p + a- + 1) 8 P ,,, , ;,, 



and a cubinvariant 



V = (x + M + i) e^e; - pe.e;,,,, ; 



from which 



Now the right-hand side is the product of a quadrinvariant and a cubinvariant, and 

 therefore J is composite. Similarly, for (6) we take 



and 



e,, 9 = pe,e p , l - (z p + 2) e p> , e; ; 



from which 



The right-hand side, as before, is the product of a quadrinvariant and a cubinvariant ; 

 and therefore J l is composite. 



The last method may be applied to (c) also, and leads to a similar result ; for, taking 



. , 2 - (2<r + 2) e^.e:, 



we have 



<re,P - (Ip + 2) P , ,9,., = (2<r + 2) e,,, {(2p + 2) 8,. ,8^ - (78,8,, ,}, 



i.e., the product of a quadrinvariant and a cubinvariant. Hence P is composite, and 

 therefore from (c) no proper quartinvariants arise. 



