388 
PROFESSOR 0. H. DARWIN ON FIGURES OF 
The third term arises from t — k, r — i — 2; t = k — 1, r = i — 1 ; t = k — 2, 
r — i, and we find— 
Third term — ~(k 1) (i -f- 1) j- — 
A symmetrical form for further terms may be obtained by writing (31) first wfith 
i before k and then with k before i, and taking half the sum of the two results. In 
computing these coefficients it is a useful check to compute from both nnsy mm etrical 
forms, when the identity of results verifies the computation. 
The following Tables have been computed from (30) and (31). The numbers are the 
coefficients of the quantities at the heads of the columns for the values of k and i 
written in the first column. The series (k, y) is terminable with /T, and the series 
\_k, i, y] is terminable with 
In [k, i, y] the coefficients have only been computed as far as /3 6 , so that the last 
which is given completely is [2, 4, y] ; however, with such values of /3 as we require, 
the series are carried far enough to give numerical results with sufficient accuracy. 
- 1) + k (k - 1) 
2 ! 
, hk (i + 2) (k + 2) , ik (i + 1c + I s ) 
H-auw-H— 
2! 3! 
2 
J8* 
Table of ( k, y). 
Log (1 + /3) 
+ P 
+ /3= 
+ (3 3 
+ /3 4 
+ /3 5 
it = 2 
3 
3 
1 
2 
.. 
s-* 
II 
CO 
4 
6 
2 
1 
3 
• . 
7c = 4 
5 
10 
5 
11 
1 3 
1 
4 
• • 
7c = 5 
6 
15 
10 
5 
H 
JL 
5 
Table of [&, i, y]. 
Log- (1 + (3) 
+ P 
+ /3» 
+ /3 3 
+ P* 
+ (3 5 
+ |3 6 
+ P 7 
7c = 2, i = 2 
9 
27 
184 
8 
14 
# # 
.. 
7c = 2, t = 3 
12 
48 
46 
31 
12 
2 
. . 
. . 
c** 1 
II 
to 
(S'. 
II 
15 
75 
924 
85 
501 
17 
2- 
7b = 2, i = 5 
18 
108 
163 
190 
1514 
m 
23 
Ac. 
A = 3, i = 3 
16 
84 
108 
103 
63 
22 
H 
• • 
s- 
II 
M 
c*. 
II 
kf^ 
20 
130 
210 
260 
219 
118 
36| 
Ac. 
7c = 3, 1 = 5 
24 
186 
362 
552 
594 
4344 
206 
Ac. 
II 
-S. 
II 
25 • 
200 
400 
625 
687| 
514 
2484 
Ac. 
7c = 4, i = 5 
30 
285 
680 
1285 
17504 
1681 
1110 
Ac. 
h = 5, i = 5 
36 
405 
1145 
2585 
4272 
5098-f 
4345 
Ac. 
