27 
developement of exponential functions , &c. 
(one of the most important). Unfortunately, his method 
turns upon an artifice which, although remarkably ingenious, 
fails to afford us any satisfaction except in this particular case; 
and I am not aware that his researches have since extended 
beyond it. The essay of Dr. Brinkley (the only author I 
have met with who has attempted the general problem) goes 
to the bottom of the difficulty, and leads to a formula which, 
considering the complex nature of the subject, must be 
allowed to be far more simple than could have been ex- 
pected. It is often, however, advantageous to undertake the 
solution of the same problem by different methods. The 
excellent geometer I have mentioned, has adopted one which 
appears at first sight very inartificial. It consists in expanding 
the second member of the equation (a) reduced to the form 
by the well-known theorem for raising a multinomial to the 
n th power. The difficulties and apparent obstacles which this 
method presents, he has overcome or eluded by a singularly 
acute discussion of the combinations of the various numerical 
coefficients and their powers. But it is obvious that this 
method, applied to the more general equation ( b ), would lead 
into details of extreme complexity. This consideration in- 
duced me to begin with that equation, regarding the other as 
a particular case of it ; and I have thus arrived at a general 
and highly interesting formula (equation (2) of the following 
pages) hitherto, I believe, totally unnoticed, and which in the 
particular case of the equation ( a ), when n is a positive 
integer, affords precisely the same result as Dr. Brinkley 
has given : when n, however, is negative, it yields an expres- 
E 2 
