IN A RESISTING MEDIUM. 
137 
The following table, extracted from Legendre^s u Traite des Fonc- 
tions Elliptiques,” Tom. II., p. 305, Table IX., gives the number of 
degrees in an angle whose circular measure is <£, corresponding to the 
argument x, the modulus being sin 15° :— 
<*> 
X 
<P 
X 
<t> 
X 
o 
O 
o 
0 
0-00000 
31 
0-54273 
61 
1-07566 
1 
01745 
32 
56035 
62 
09358 
2 
03491 
33 
57797 
63 
11151 
3 
05236 
34 
59561 
64 
12945 
4 
06982 
35 
61325 
65 
14740 
5 
08727 
36 
63090 
66 
16536 
6 
10473 
37 
64857 
67 
18333 
7 
12219 
38 
66624 
68 
20130 
8 
13966 
39 
68393 
69 
21928 
9 
15712 
40 
70162 
70 
23727 
10 
17459 
41 
71933 
71 
25527 
11 
19206 
42 
73704 
72 
27328 
12 
20954 
43 
75477 
73 
29127 
13 
22702 
44 
77251 
74 
30930 
14 
24551 
45 
79025 
75 
32733 
15 
26200 
46 
80801 
76 
34635 
16 
27949 
47 
82578 
77 
36339 
17 
29699 
48 
84356 
78 
38143 
18 
31450 
49 
86135 
79 
39947 
19 
33201 
50 
87915 
80 
41752 
20 
34953 
51 
89697 
81 
43557 
21 
36706 
52 
91479 
82 
45362 
22 
38459 
53 
93262 
83 
47168 
23 
40213 
54 
95047 
84 
48974 
24 
41968 
55 
96832 
85 
50781 
25 
43723 
56 
98618 
86 
52587 
26 
45479 
57 
1-00406 
87 
54394 
27 
47236 
58 
02194 
88 
56200 
28 
48994 
59 
03984 
89 
58007 
29 
30 
50753 
52513 
60 
05774 
90 
1-59814 
= K 
This is all the tabular matter required for the calculation of 
Mr. BashfortlTs X y ; for 
z — b — h >/3 
Cn X ~ * -fi + Is /3 ’ 
X_Cn * + 4(^3-1)’ 
8* _ 1 (4 + 4& 2 )i P^ — ( s/_ 3 +1) (1 + ( a — p) 
g (4 + a 2 )o (4 + 4« 2 T Cn (4 + 4« 2 TPt + (\/3 — l)(4+« 2 )^(a— -p) * 
or 
or 
giving x in terms oi p ; and Mr. BashfortlTs “X^ 
p _ 9* 
Again, 
_ X 2 
i — 5 3 “ X 2 ’ 
y- 3 « a ^ 
4 + 4« 2 A 2 * 
or 
