GENERAL THEORY 
197 
Then 1 /P can he developed in a power series 
c Q + cpc + cpc 2 + 
valid in 33. The first coefficient c 0 = 
For 
P a 0 + $ a o ]_ i 
a o 
if, q q 2 æ , 
Ct(\ l Ctf\ Ctn €t\ 
for all x in 33. We have now only to apply 161, l. 
3. Suppose p = a mX ~ + a ^ x ^+ ... « m¥= 0. 
To reduce this case to the former, we remark that 
P = x m Q 
Q = a m + a m+1 x + ... 
Then 111 
where 
P x m Q 
But 1 /Q has been treated in 2. 
164. 1. Although the reasoning in 161 affords us a method of 
determining the coefficients in the development of the quotient of 
two power series, there is a more expeditious method applicable 
also to many other problems, called the method of undetermined 
coefficients. It rests on the hypothesis that/(a;) can be developed 
in a power series in a certain interval about some point, let us say 
the origin. Having assured ourselves on this head, we set 
f(x) = a 0 + a x x + a^x 2 + ••• 
where the a's are undetermined coefficients. We seek enough 
relations between the a’s to determine as many of them as we 
need. The spirit of the method will be readily grasped by the 
aid of the following examples. 
Let us first prove the following theorem, which will sometimes 
shorten our labor.