426 MB. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



and hence II can be expressed in terms of the invariants which have been retained as 

 fundamental and of proper derived invariants. Thus, in the case of the invariantive 

 combination suggested by the associate variable of the second rank, only composite 

 invariants are obtained ; the result is general for all the invariantive combinations 

 thus suggested. 



Combinations of the Associate Variables. 



61. Consider now the complete set of dependent variables, viz., the original 

 variable y and the n 2 associate variables, as a fundamental set. Any one of them 

 satisfies a linear differential equation of determinate order ; and with it, as an original 

 dependent variable, there will be associated a number of new dependent variables, in 

 number 2 less than the order of the equation, and functionally derived from it in the 

 same manner as the preceding have been derived from y. 



Taking a few simple cases, consider first that of the associate of the first rank 

 and let 



^12 = 2/i2/2 - 2/22/1 ; V M = y z y\ - 



so that v 12 , v 34 ,, v B6 , are particular solutions of the equation whose dependent variable 

 t z is the associate of the first rank. One of the set of variables, associate of the first 

 rank with 2 , is 



2/i2/2 - 2/2/K 2/1/2 - 2/2/1 



2/32/4 - 2/42/3, 2/3/4 - 2/4/3 



= 2/2 



2/1 > ?/s , 2/4 



2/3 > 



y\, 



2/a > 2/3 , 2/4 



2/2, 2/3, 2/4 



and is, therefore, expressible in terms of associates of the original variable y. Again, 

 one of the set of variables, associate of the second rank with t 2 , is 



r 



'66 



'34 



12, 



S/lS/2 - 2/22/1, 2/32/4 - 2/42/3, 2/52/6 - !/62/5 



2/1/2 - 2/2/1, 2/3/4 - 2/4/3> 2/5/0 ~ 2/6/5 



2/1/2 - 



. 2/ 8 2/ U 4 - 2/4/3, 2/6/6 - 2/62/ U 5 



2/12/2 - 2/22/1, 2/32/4 - 2/4/3, 2/52/6 - 2/62/5 

 2/1/2 - 2/2/1, 2/3/4 - 2/4/3, 2/5/6 ~ 2/6/6 

 2A/2 - 2/2/1, 2/3/4 - 2/4/3, 2/6/6 ~ 2/6/5 



- 2/22/3 V 1466 ~ 



56 



2/22/4 ^1356 ~ 



