30 Mr Herschel on the application a new mode of Analysis 
^ 2 + 
1 
"Y+a 
^{(0 + S)'?L(0-i)"} 
{t» 
2j'(2iV — l)(S.r — 2) 
1.2,3 
0^ 
&c 
} 
But we have seen that ^ ^ __ ^ 
2 -f A j gx—l 
(by the transformation E) ; and for cc writing a? — 1, — 2, &c. 
and substituting in the above expression, we obtain for our final 
value of S 
2a: -}- 1 
{(-!)* 
+ 
1 
B, 
1.2... .2^7 
^x{%x — 1).22^— 3 (22^ — 2_ 1) ^ ^ 
-^2a;-5 + 
which is the expression referred to. But the value in (G), with- 
out any transformation, is at once simpler as a symbolic repre- 
sentation, and easier to reduce into numbers. 
(9.) Let us now consider the series 
^ (0 - Ao. /(O) + A,. /(O L A,. / (2 0 + &C. 
and endeavour to discover, by direct investigation, all the pos- 
sible corresponding forms of the co-efficients Aq, A^, &c. and 
the function which shall render (p (^) equal to a rational in- 
tegral function of &, or 
^ ~ a “b • • • • h& 
For this purpose we must develope (p (f) in powers of and the 
co-efficient of any indeterminate power, being equated to zero 
with the annexed condition a’> ^ will furnish the condition 
which resolves the problem. Suppose then 
F (^) = Aq -f A;^^ -f Ag + &c. 
/ (f) = «o + a^t^ &c. 
and we shall readily see that the co-efficient of being the sum 
of those in each of its separate terms, will be 
ffl^x|A<,.C* + A,.n + A,.2^+&c.| 
Now this, as we have already seen, is expressed by 
a^* F (1 -f a) 0% hence we must have 
«^xF(1 + a)0^ = 0; [x>n]. 
There are three ways of satisfying this equation, 1st, By sup- 
