320 NEW SERIES for the 
52. Ler w denote any number greater than unity, and p and 
n any two whole pofitive numbers ; then, by a known formula 
uv 
ye — IT = 
—tfutytepor.. te}, 
fopopopot... tor fs 
therefore, dividing each fide of the firft of thefe equations by 
the correfponding fide of the fecond, we get 
v—r i vtv+tu+u...+ 
vi v+ut+utu... fu 
Now, v being by hypothefis greater than unity, the fraction 
on the right hand fide of this equation is lefs than this other 
fraction 
vo + or t+ve+v?...+v? (to pterms) _ pv? 
aes Cr EE rh. 
becaufe it has manifeftly a lefs numerator, and at the fame time 
a greater denominator. The fame fraction is, however, greater 
than this fraction 
rti1t+i1+i1.--+1 (topterms) _ 
Se eer ee Garten 
voto +o... +0" (tom terms) — ip 
becaufe it has a greater numerator, and a lefs denominator. 
Therefore, 
we—T ye pe , 
< p = > p ? 
Peo T n U—TI nv" 
and hence, dividing the firft of thefe expreflions by v¢, and mul- 
tiplying the fecond by v", 
yb —"T vo" (vt — 1) 
no w(t —t1)’ yeaah (a). 
53. PuTTING 
