44 
Mr. Herschel on the 
S denoting that the sum of all the possible values of the 
expression within the brackets is to be taken ; a, (1, .... v, 
(whose number is n) varying through all integer values, 
separately, from o to co . Now the several factors which 
compose this expression are respectively, the coefficients of 
in the developements of// (£), (/£)*, (/£)/ . . . . 
(t Let these coefficients be represented by H a » “K#* ^K y , 
. . . . and we shall find 
/+* (0 = s { H. . . % “K, . f } ; (33) 
If for instance, / (t) = e t == log.— i /), we have 
H = /(* 4 A )o« . 
« 12..,. a 
*K, 
(3 
1.2 .... /3 
,, &C. 
whence we obtain 
/io g .-» ( t) = (m) 
To take another example, let us suppose the developement 
of /+"(0 were required, where / (/) = e f _ i. In this case 
'equation (/) becomes simply 
/+(0=/( A)^ 
and the formula in (32) gives 
f^ n [t) =/(A)s°- a ' + 0 '- A“ + ....o(»~0.f 
In this case also (33) becomes 
• A* ■ A g Al£ z ,1. 
which gives, if/ (2) = t, 
,J, ■( M — .«? j A / X A g . A** o’ f , I 
V ' ( I....|3 x I .... V J 
fV (0 = 5 
Now Ao 0 = 1, and if, for the sake of symmetry we write 
a, /3, ^ instead of j3, 7, . . . . we shall have 
