140 Sir James Cockle on Primary Forms, 



where 



Q=M + l-A + E-f2AE=,E 2 -(A-E) 2 . 



Next differentiate (19), replace -j- by y, and take the criticoid of 

 the result. We have 



^| + {M + l-(A + l)Acot 2 ^-(E-l)Etan 2 ^ 



-2A + 2E}2/=0, (20) 



of which a conjugate is 



^+2{(A+l)cot^+(E-l)tan^}^-hQ l2 / = 0, (21) 



where 



Q 1 = M + 1-2A + 2E-(A + 1) + (E-1) + 2(A + 1)(E-1) 



= Q-4A + 4E-4. 

 Hence 



Q, = ,E 2 -(A-E + 2) 2 ; (22) 



and, if neither A nor E is an integer, we shall, after performing 

 this process n times, transform (19) into 



3 +2P »2 +c ^=°' <w 



where 



P w = (A + n) cot x + (E — n) tan x } 

 and 



Q n =^ 2 -(A-E + 2rc) 2 . 



But Q n will vanish if iE + (A— E) is an even integer. Now 

 the two values of A are connected by A 1 H-A 2 =1; so that if, for 

 instance, M + A l ^ E be even, then M— k x — E will be odd; 

 and the condition coincides with one obained from Boole's 

 process. 



12. For clearness I have supposed that neither A nor E is 

 entire. But if both or either be so the process is not stopped. 

 The identities c 2 + c=(c + l) 2 — (c+l)andc 2 — c=(c — l) 2 + c— 1 

 give us a choice of conjugates. Let E = w, then tan x disappears 

 from (20) and its conjugate ; but, since c= — 1 satisfies c 2 -f c=0 3 

 it may be made to reappear. When A and E are both entire, 

 P n may be made to vanish ; and Q n is always a constant. I 

 believe that the results of this process are coextensive with the 

 regular results of that of Boole, and that it applies to such forms 

 as 



^+(tano: + 2Ccot#)^ + Kcot 2 *.y, 

 dx z ax 



