451 
of Edinburgh, Session 1879-80. 
I oK) = l + ^ + g + &0 . 
(37), 
tlie constant factor being taken so as to make I 0 (0) = 1. 
Stokes * investigated the relation between E and D to make w — 0 
when r = oo and found it to be 
E/D = log8 + 7r-ir'i= +2-079442- 1*963510- -11593 1 
or, to 20 places, E/D = -11593 15156 58412 44881 j ( 38 )‘ 
Hence, and by convenient assumption for constant factor, 
1 0 (mr) = log 
mr 
1 + 
n 2 r 2 m 4 r 4 
2^ + 2?.P + &C ' 
Yi 2n*2 TV) 4/v»4 
|L(S 1 + -11593) + j£L<S, 
•11593) + &c. 
(39). 
It is to be remarked that the series in (36) and (39) are conver- 
gent however great be mr ; though for values of mr exceeding 6 or 
7 the semi-convergent expressions t will give the values of the 
functions nearly enough for most practical purposes, with much less 
arithmetical labour. 
Erom (37) and (39) we find by differentiation 
x / x mr , m°r , no / , « 
IiM = ^+2^ + 2T 4 T6 + &c- 
-r: / x 1 , 3m 2 r 2 , 5 mV 4 , « 
I.K) = T + ^r ¥ + 2 -r S 6 + &c- 
3^.3 
m°r 
5->.5 
(40). 
* “ On the Effect of Internal Friction on the Motion of Pendulums,” equa- 
tions (93) and (106 ). — Cambridge Phil. Trans., Dec. 1850. 
P.S . — I am informed by Mr J. W. L. Glaisher that Gauss, in section 32 of 
ci /3 
his “ Disqusitiones Generales circa seriem infinitam 1 + x + (Opera, 
vol. iii. p. 155), gives the value of -7r~h , '|, or i n his notation, to 23 
places as follows : — 
1-96351 00260 21423 47944 099. 
Thus it appears that the last figure in Stokes’ result (106) ought, as in the 
text, to be 0 instead of 2. In Callet’s Tables we find 
log e 8 = 2-07944 15416 79835 92825, 
and subtracting the former number from this we have the value of E to 20 
places given in the text, 
t Stokes, ibid. 
VOL. X. 3 I 
