sums of several classes of infinite series. 26*7 
only a particular case of a much more general principle, which 
is of use in discovering certain values of the variable, in which 
a series admits of summation, but which generally is not 
expressible in finite terms. The principle is as follows : let 
K denote any operation, such as integration, either with respect 
to differential or finite differences, or any other operation, pro- 
vided only K( X+ T) = KX + KT Now let 
K\J/i37 A^x — J— -j- A^.£ 3 -|— 
Put ax, a*x , . . . a n ~~ 'x for x, and the results will be 
JCvJ/a:# = A x aX -J- A 2 a t r 2 -J- A ^aX 3 -j-’ 
A t a 2 X -}- A 2 oiOC 2 -J- A^a 3 x 3 -|“ 
&c. &C. 
= A x a n " l x + A 2 « m “V+ A 3 ^“ , a? 3 + 
By adding all these together, we shall have + fa* -f- 
V-z + See. \! >a n \r j equal to a series whose general term is 
+ aX -j- aX + Sec. -J- «” ‘a?! . Now supposing we can- 
not perform the operation denoted by K on the function fa, 
yet if t|/ is of such a form that -\x -{- fax -f- Sec. -f- -fycc n ~ l x is 
equal to a function on which the operation K can be executed, 
then calling this new function 4^, we shall have 
Kifqa? = SA f ^x' -J— ax'-f —..ot 1 I cj?* 
And if a is such a function that a n x = x, a great variety of 
forms for may be found, which will satisfy that condition. 
Now let x be determined by the equation x = ax, and r being 
any root of this, we have r — ar=za 2 r= Sc = a n ~ l r. 
Consequently our equation becomes 
K^o; = nSAfzss n^Ar-\~Ar 2 fa A f 3m h A^ 4 -j“ &c.| 
