. 
[900 
901] 
49 
g so-called lecture 34, 
3 that of the functions 
y", ..., in regard to x 
ed by the interchange 
than that imposed by 
e theory is accordingly 
surprised to reflect on 
f a century. It is now 
och of it. These thirty 
,nts, might be compared 
if Algebra had attained 
•ee hand' to develop his 
■ Formerly, it consisted 
a which every properly - 
expected to suggest the 
canvas.” And, indeed, 
ily in these lectures by 
hers,—Elliott, Hammond, 
Hamiltonian numbers : 
l Hammond jointly, two 
that of the series of 
N. H. Hamilton in his 
nethod. A formula for 
)tained by Sylvester, but 
discovered by Hammond, 
foregoing numbers, each 
a 
21, 
serve the paradox, t=\, 
i 
2- 
tester’s achievements in 
in the foregoing very 
901. 
NOTE ON THE SUMS OF TWO SERIES. 
[From the Messenger of Mathematics, vol. xix. (1890), pp. 29—31.] 
I CONSIDER the two series 
and 
S = 
Si = 
+ 
+ 
1 + e™ 3 (1 + e 3 ™) 5 (1 + e 5 ™) 
+ ..., 
+ 
2 + 7ra 3(2 + 37ra) 5 (2 + hira) 
where a is real, positive, and indefinitely small; these would at first sight appear 
to be equal to each other, but this is not in fact the case. 
Taking first the series S u putting therein 7ra = 2x, this is 
2/8,= 
1 
+ 
+ 
+.... 
1 + x ' 3 (1 + 3x) T 5(l + 5x) 
Now we have, (Legendre, Théorie des Fonctions Elliptiques, t. il. p. 438), 
ïTÿ + ïf+ÿ) + 3(3+7) + ""' = ° + é ‘ 0g r (y+11 
where G is Euler’s constant, ='577...; and if y be real, positive, and very large, 
then 
r {y + 1) = V(2tt) y y+h e^ + ^y + " ; 
whence, differentiating the logarithm and neglecting the terms which contain negative 
powers of y, then the value is = C + log y ; hence, writing y = ^, we obtain 
1 
l + x + 2(l+2x) + 3(l+3x) + logX - 
C. XIII.