Mr Herschel on the application of a new mode of Analysis 
we shall find by our general formula (B) 
R (log l^A)0^=p,; n (log 1 + A) 0" = l.9.p^, &c. 
wherefore, 
<p($)=za^p^ + 1.9. a^p,, ^2-j. .,..1.2 nxa^Pni"'. (H) 
because in this case n must be an even number. 
We may know at once whether any assigned form of F (^) satis- . 
fies the above condition, without actually determining the form 
of 4^, by regarding the equation- 
^(i)_4(l)^F(i)_R(log^) 
as a fundamental equation for finding 4" a^nd enquiring the 
condition of its possibility, which will appear to be 
F(0 + F(i)=R(log0 + R(-logi>^ (K) 
and if on substitution in the first member it be found that there 
does not result either zero, or an even rational integral function 
of log the form assigned to F is contradictory. If, on the 
other hand, such a function does result, it is satisfactoiy ; and 
R (log if) may immediately be had by taking half this result. 
(11.) In exactly the same manner, if the even values of A'^ 
vanish, and the odd ones of F (1 + A) 0% we have f (f) =any 
function developable in odd positive powers of ^ ; and 
F (^) = 4" (0 + 4 " C.~i^ ^ 
which gives for the condition of possibility 
F (0 - F(i)=R (log O-R (-logo? (L) 
any odd rational integral function of log t, the half of which 
being taken, gives R (log if) and of course the co-efficients 
&c. which found, we have (n being odd in this case) 
4,(«) = 1. tti-pi + 1.2.a.a3^3 + .... 1.2.... ra (M) 
(12.) Other cases, though of less analytical neatness, might 
easily be devised ; but it will suffice to exemplify these by one 
instance. Let then 
F (0 =”L (1 + t)-- -+ ^ &c. 
r s” 3" 
and we have ' 
^ an ' on 
