Equations of the First Order. 



123 



determining apriori either the proper soluble form to begin with, 

 or the proper transformation to apply to it. Its utility consists 

 rather in enabling us to make an extensive repertory of inte- 

 grable forms, which may be consulted in order to ascertain 

 whether any proposed equation lies among them. 



The first object must be to find a convenient set of pairs of 

 correlated forms. The more simple they are in form, the more 

 useful they will be likely to be for the purpose of transforma- 

 tion. The following Table contains all that I have met with 

 or discovered of a simple character. The list may be extended 

 indefinitely by combining them one with another; but they 

 soon assume a complex form. 



Correlated Functions. 



No. 



u. 



V, 



w. 



1 

 2 

 3 



X 



ax+py 

 xp + (a — l)y 



y 



yia+prf 

 px a 



P 



w a-l 



4 



x . 



Ja-Mrt* 



(1+ap*)-* 



5 





4)' 



4)' 



6 



? 



*P + «(*-j) 



*+f 



7 



p(x+ l)-y 



x K ' 



(1+tf 2 )"* 



8 



p n ~ 1 + (n — l)ax 



p n + nay 



n 



9 



x n ~ l + (n — l)ap 



x n + na(px~y) 



n 



r x 



n—1 



10 



P 



cj)p + a(xp—y) 



<f>'p + ax 



In this Table we may in each case transpose u and v, and sub- 

 stitute for w its reciprocal, and thus we obtain a new set of cor- 

 related functions. Or we may invert each set by finding x } y, 

 and p from the system 



M = X, V = Y, 20 = P, 



and considering the new forms so obtained as new sets of corre- 

 lated functions. Thus No. 2 in the Table produces (restoring 



