404 
SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
(16.) These relations premised, let it be required to express the sum of x terms of 
the series 
^(a+^) + (p(a+2J) + ^(a+a^&)=S^. 
Developing the several terms, we find 
S^=(p(a){l + ] + X terms} 
+}.?'(a){l+2+3+ X} 
+~f"{a){V+2'+3^+ ....x'}+ &.C. 
Substituting then for the series of powers of 1, 2, 3, &c. their values as given by 
equation (11.), and separating the symbols of operation from that of quantity, we get 
^(a).O'’— (p'(a).^.0’ + ^"(a).^.0"— = (p(a — 6.0). (28.) 
( 17 -) If we use in like manner equation (12.), it gives 
S^=(l + A)-- -'^^^ — = — -<p{a+b-\-b.O), . . . (29.) 
by employing the transformation of equation (3.), in which x= 1, /(O) =(p(a+Z>.0). 
Hence also, if 
— ?’(®)4'<P(^+^)4‘ •• ..^{ci-\-x — 1 .b), 
we find in like manner 
S.= (l'^y V (a+^,-0) (30.) 
(18.) Let it next be required to find the sum of the series 
to y terms, where y= = the integer part of the quotient of an independent index 
number x, divided by any given number s. By equation (28.) we have 
Now since 
S,= ' + 
X X — 1 X — s+ 1 
•?.-!+ — ^ 
If we put 
s~s‘| 1 ....(■?— l)-'?^- 
■s+] 
P—-~S’ ^ = 7{0--S.r+l -y.r-l+ &C.|, 
we get by equation (9.), 
S^= ^ ~ ^ ^ ■ -^(a — ^ . 0) + ^ ^ ^'^' <p{a-pb-b.O). 
