1 66 Mr. Ggmpertz on Series which may be summed 
whole number, that these two values must then be equal to 
each other, and that &c. containing a finite 
number of terms must be then equal to a-\-br-\-cr' &c. con- 
taining a finite number of terms, and consequently the highest 
power of r and its coefficient must be the same in both series, 
otherwise by increasing r by the same in both, one side of 
the equation would become greater than the other, which is 
absurd ; consequently the highest power of r and its coefficient 
is the same in both, and will destroy each other, and conse- 
quently the next highest powers of r and likewise their 
coefficients must be the same with each other, and will there- 
fore be destroyed, &c. Hence the powers of r and their 
respective coefficients being the same in both, the expressions 
themselves must be the same in every respect, whether r be 
a whole number or not. 
Hence we have not only given two different means of sum- 
ming the series p m — r .p-\~q\ m &c. (m being a whole positive 
number ) whatever r may be, which indeed was not our chief 
object, but we have likewise proved that the series sine of 
pz—r sine of p+q.z & and the series cos. of 
1 z cos. of \qz\ 
pz—r cos. of P4-O.Z &c.= whatever r may be, the 
* 2, COS. of ~qz\ r 
same as Landen finds. 
Cor. ii. Because these two series are equally true, whatever 
p may be, if for p we write qr — p throughout, in the first we 
shall have, sine of qr— p . z—r sine of r— |-i . q — p . 2+ r 
r + i 
sine of r-{- 2 .q—p.z &c.= 
sine of iqr — p.z 
7 of 
2 COS 
sine of p — \q r z 
2 cos. of \q \ r 
Consequently sine of qr—p.z—r sine of r-^iq—p .%-j- 
