NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 407 
&c., we have 
+ 0 . A') = - y AO . ^ ■ 
5—1.25—1 . 5—1 . 
AW 5 AO^ 
1.2.3 1.2-^-^^ 
vI/"(5) 
)-V. 
-( 
5— 1.25— 1 .35—1 
1 . 2 . 3. 4 
.AW 
S- 1-2S-1 A203,«-:I ,.2 A03^ 
1.2.3 ® 1.2 y ' 1.2.3 
(36.) 
+&c.j. 
Hence if we put 
Y(.s) = 0 . + -4/1 (.s) . 1 + ^2 (.s) . + ■^s-i{s)s^-s+i, 
and denote by &c. the differential coefficients of regarding the 
discontinuous functions s^,s^_^, See. as mcapahle of differentiation, we shall have, finally, 
5[ ^ 1.2 1 1.2.3 ^ 1.2’ • ('1.2 
I 
J 
f5-1...35-l 
“L 1--4 
+ &C.| . . 
AW- ^ 1^2% ^ -^-AW 
5-1 
2 2'* • ^0 j 2 , 2 . 3 
(37.) 
(23.) As regards s, in every part of the following investigation it will be regarded 
as a mere given integer number, so that the coefficients in s will come to be calculated 
in absolute numbers and need give no further concern. It is otherwise with those 
in X, which mix themselves up with the x contained in ©( 47 + 5 — 1) and its differential 
coefficients in a way requiring special examination, as functions of an independent 
variable. Let us therefore consider the term multiplied by f^-j ^(7r+^— 1) in the 
development of <f(j7+6— 1 — .s.O). This term will be expressed in Arbogast’s nota- 
tion by 
Qn 
and the corresponding term in X will be 
The development of this in factorials x, Tr-f-.?, x-\-2s, &c., and functions <p, ®', <p'’, &c., 
is accomplished in equation (35.), but if we would effect it in powers of x we must 
proceed as follows : — 
Suppose 6=2-7182818, &c. Then we have 
l-(l+A)-T ^._ 1^£2 A!Ao. 
£_ log(l+A) x‘ {log(l + A)}’= 
l.s’ A ^ 1.2.s‘^* A 
{log(l+A)}3 
1.2.3' A 
0”— &C, 
