770 PEOrESSOE BOOLE ON THE COMPAEISON OF TEANSCENDENTS, WITH 
But 
log (ot+^COS 0 . .) = log a^+log w- 
■ cos 5 
+ &c. 
log(2m^+'i;^ — . .)= log 2m? 
Substituting, we have 
(^-cos 6 . .^|log^+logi;-log 2nf+ • -j’ 
1 . 
wherein the coefficient of - in the product is 
cos 0 V 
(13.) 
In the second place, developing the function in descending powers of x, we have 
>/ X* — 2'm?C(f' cos 0-{-m*=.C(f ' — cos d. See., 
which on substitution gives 
TO^cosS \ — ir2 + z;a? + m^(l +cosfl) 
0 
x^- 
rr? cos 9 
.)log- 
x‘^-\-vx-^ m^{\ — cos 9) 
2v 
f ^ n? cos 9 \ / 2v \ 
= fi 
1 . 
wherein the coefficient of - is — 2v. Hence, changing its sign and adding the result to 
(13.), we have 
^ x'^— cos 9 + , 3 , cos 9 , ^ 
^ (7^=2 hC, 
which agrees with (8.), art. 17. 
It is, however, very much easier in the above problem, and perhaps in most others, to 
apply at once the fundamental theorem of transformation, as akeady exemplified in its 
solution. 
24. Abel’s theorem is of course included in that of Art. 23. To deduce it, we must 
observe that its object is to determine the value of the expression 
J{x-a) ^/,p^{x)<p2{xy 
the simultaneous values of x being connected by the follot^fing equation, — 
, /^) _ gp + 
V ^i(^) CQ-j-CiX..+C^X» ' 
To compare with the general theorem we must therefore write (1.) in the form 
./W , /fd^ 
^J{x-a)<p^{x)V <p^{x) 
Hence we must make in (1.), 
/(^) 
dx. 
