1906-7.] Periods of the Elliptic Functions of Weierstrass. 361 
The march of these series, computed both accurately and by the 
approximate formulae, is shown in Table I. 
Table I. 
‘1 
io g F (- 6 -§> 2 > sin 2 D- 
By Approx. 
Formula. 
logP (g>I’ 2 ’ sin 2 l) 
By Approx. 
Formula, 
0 
0 
00000000 
00 
0-0000000 
00 
10 
9-9996784 
84 
0-0002297 
97 
20 
9-9987132 
32 
0-0009184 
84 
30 
9-9971030 
30 
0-0020653 
53 
40 
9-9948459 
59 
0-0036690 
90 
50 
9-9919398 
98 
0-0057269 
69 
60 
9-9883830 
30 
0-0082349 
48 
70 
9-9841755 
54 
0-0111868 
67 
80 
9-9793202 
00 
0-0145732 
32 
90 
9-9738259 
59 
0-0183800 
00 
180 
99130246 
0-0591526 
The following relations hold : — 
7 4 
1 
“6 
^ 2^ 
5 6 ’ ) 
II 
Jyj 
2 1 5 
v 6 ’ 6 
y (- — 
7 
,2,1) 
\ f (— , 
^ 2 
’ 6 
’ 2 7 
\ 6 5 
6 ’ ’ 
■— , 2 , cos 2 7l\f| 
( 1 
— . 2 . sin 2 7L\ 
V 6 6 2 7' 
V6 ’ 
6 7 7 
2 7 
n 108 2 
70 ' 7T 
cot 2 -h- 
Also 
W - 14 , 2 , cos^j + , f , 2 , cos^j - 10 cos^I^ , ^ , 
i 4 • 2 ■ sin y) + Kl > i • 2 ’ sin y) - 10 sin y F (y ■ t • 
7F( 
2 7 
35 
7 r ’ 
1 
5 
, cos 2 -h-) 
2 . 
6 ’ 
6 
2/ 
1 
5 
,2, 
, sin 2 -h) 
,6 ’ 
6 
27 
— 5 — F f -L , -P-,2, sin 2 -^ = 
9 4 V 6 ’ 6 5 5 2 7 
cos 2 -^- 
_2 108 
TT • o fc 8 
sm y 
7F( 
1 
1 4 • 2 1 sin 4) + 5 cos ‘44 i ’ 2 ’ sin2 2 -) 
K _/ 1 fi 
A__ F fl 5 A, 2 5 
^ 2*3 5 6 
COS 2 -^- 
