242 Mr. Horner on [April, 



" though," adds he, " to demonstrate this in any case but when 

 « = 2, seems a matter of some difficulty." 



I submit, that the converse of this, perhaps, is Ukely to be 

 correct, viz. that all real solutions of vp" x = x are comprised in 

 the formula 



il/ o; = <p -r- — 



^ ^ c ^ a <f X 



which, as will readily appear, resolves itself into two distinct 

 and very simple formulae, according as ?« is = 2 or > 2. 



In fact, by making a, c, d, vanish, and putting b = (1)", Mr. 

 Herschel's formula is immediately obtained ; but as it does not 

 appear to me that this can be reduced to a real form when ii > 2, 

 and it has moreover, in this instance, been obtained by sacrific- 

 in^y the arbitrary character of the constants, we will pass on to 

 another mode. 



10. Not to multiply symbols in an easy inquiry, the reader 

 will recollect that the values of 9 and \J/ change, from step to 

 step of the process. 



b 



Make then <px = x + -, and we have 



^^ = ^~\ij+l^dfx) ....(29) 



Where (1st) if 6 + c == 0, we have, putting — ^5 — =C, 



t -. C 



\l/.r = a ' — , 



^ ~ <px' 



which is obviously a solution of v}/'- x = x. But again, put 

 <^ X — ± X V C, and we have 



4,^ = ^-'^ (30) 



for a general solution of the second order of functional circles. 

 But secondly, retaining b and c arbitrary in eq. 29, make 



<p X = -, and it becomes, putting +bc + ad=B, and b + c 



= A, 



^x=f-'^^ (31) 



^ ^ A ?F fx 



Or, putting again (p x = x V B, 

 If (p a; be again made = -, we have 



i (S2) 



