451 
of Edinburgh, Session 1879-80. 
\{mr) = 1 + -- + — + &c. 
(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—cc and found it to be 
E/D = log8 + 7r-ir'i= +2*079442- 1-963510= -11593 ) 
or, to 20 places, E/D = -11593 15156 58412 44881 J 
Hence, and by convenient assumption for constant factor, 
log 
+ ^-(8r + -1 ■ II 593) + (S 2 + -11593) + &c 
1 + 
[dr 2 
~W 
mrr 
4™4 
+iii +&c - 
(39). 
9,2 
22.42 
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. 
From (37) and (39) we find by differentiation 
T / x mr , m 3 r 3 , m 5 r 5 , p 
I l(mr ) = — + — + — — + &C. 
T / / x 1 , 3 m 2 r 2 5 m 4 r 4 „ i 
i.K) = T +^r i - + 2 — 4 jg + &c. | 
(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 
his “ Disqusitiones Generales circa seriem infinitam 1 +p~ a? + &c.,” (Opera, 
vol. iii. p. 155), gives the value of -7r~T'J, or |) in 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 he 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 
