sums of several classes of infinite series. 
251 
the sum given by this method. In this latter case, however, 
the mode of summation which I have proposed, is not well 
adapted for giving the sums of series ; its greatest advantage 
is felt when the integral or series alluded to is finite : but 
even in this case the criterion I have pointed out is not use- 
less, for it serves to except certain particular values of the 
variables, which would give incorrect results. Without this 
criterion, or without something equivalent to it, I am inclined * 
to think that the principle on which this method is founded, 
although it will probably in many cases give accurate results, 
will in others produce such as are not only numerically but 
symbolically untrue. It is worthy of remark, that the me- 
thod. of expanding horizontally and summing vertically , in many 
instances, gives precisely the same formulas as the direct pro- 
cess of integration ; yet that that method attaches limitations 
to them, which are necessary to their accuracy, but which are 
not indicated by the method last mentioned. 
Before I proceed to explain these two processes, it will be 
convenient to prove that the values of all series of the forms 
Ax 
(sin 0) m 
(cos 0)" 
+ Ax 2 
(sin 20) m 
(cos 20)” 
-}- Ax 3 
(sin 38)'” 
(cos 30)" 
■j* &C. 
A (cos _0>” 
^ (sin 0)” 
+ Ax 
2 
2 (COS 20)” 1 
(sin 20)” 
+ Ax 3 
3 
(cos 38)'" 
(sin 38)” 
+ &e. 
depend on series of the form 
(cos 6) 
a + A 
(COS 20)” 
&C. 
A x 
sin 0 
(cos 0)” 
-{- Ax 2 
2 
sin 20 
(cos 20)” 
+ &C. 
A 
* ! A ** 
(sin 0)” ‘ (sin 20)” 
~f“ &C. 
A x 
cos 0 
(sin 0)” 
Ax°- 
COS 20 
(sin 20)” 
&C. 
and 
