402 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



Proceeding in this way and remembering the analogy of the simpler cases of s, we 

 should be able to gradually reduce the highest index which occurs on the right-hand 

 side ; but the terms will become more complicated, owing to the successive differen- 

 tiations that take place. Now the form of the result to which we are to attain is 

 known, being an invariantive relation ; the successive operations earned out in the 

 way indicated always decrease the highest index on the right-hand side and will 

 never re-introduce a coefficient P already eliminated ; hence, the result attainable from 

 the foregoing starting point is unique, and we are therefore led to the conclusion : 

 There is only a single non-composite linear independent invariant for each index from 

 3 to n, and, therefore, there are in all n 2 linear invariants. 



27. But, again, as the successive steps in the gradual reduction are taken, differential 

 coefficients of Z of various orders will enter as factors with P,_ 2 , P,_ 3 , . . . and 

 differential coefficients of these, the function on the right-hand side being always an 

 integral function of Z and its derivatives. Now, for the latter, we could substitute 

 from equations like (6) and (7), and others which have not been given ; but every 

 derivative of Z can be obtained from (6) and from equations deduced from it alone by 

 differentiation. In that case there would be introduced into the terms containing 

 P,_ 2 , and Z and its derivatives factors of the form P 2 or powers of P 2 or deriva- 

 tives of P 2 , or combinations of these ; with the result that, when all the operations 

 are completed so as to leave the invariantive equation, the non-linear terms in (x) 

 will each contain at least one factor which is either P 2 or a power of P 2 or a differential 

 coefficient of P 2 . Hence each of the non-composite linear independent invariants 

 consists of two parts : 



(a) A part which is linear in the coefficients P and differential coefficients of these 

 quantities P, each term having the proper dimension -number ; 



(b) A part which is of the second and higher degrees in these quantities, each term 

 having the proper dimension-number, and every term having at least one factor which 

 is either P 2 or some derivative of P 2 . 



These general conclusions are evidently satisfied in the case of the linear invariants 

 already obtained. 



28. It will be proved immediately that the numerical coefficients iu the linear 

 part are independent of n, the order of the equation ; those of the non-linear part 

 are not independent of n, as may be seen from the special cases already discussed. 

 Hence the linear part is the same for equations of all orders not less than the index 

 of the invariant, but the non-linear part varies from one equation to another ; and 

 therefore BRIOSCHI'S remark made d propos of 3 for the cubic and quartio and of ) + 

 for the quartic "que ces formes invariantives restent les memes pour les Equations 

 differentielles d'ordre supdrieur " applies only to the linear part. 



