782 
SUMMARY OF CURRENT RESEARCHES RELATING TO 
= 0*105, 0*210, 0*315, 0*420, are given with six-figure logarithms, 
as follows : — 
sin 2 a 
Z D 
K 
0*105 
165*49 
1*23589 
0*210 
162*86 
1*74886 
0*315 
16- *47 
2*14060 
0*420 
163*13 
2*46455 
If by the help of these data the distance of union is brought in the 
usual way into the form Z = A — B ¥ + C ¥ — D ¥ } we have with 
the exactness to be attained by six-figure logarithms, 
Z D = 171 * 51 - 5 * 307 h 2 + 0 * 9765 ¥ - 0 * 05431 ¥, and similarly : 
Z c = 175*87 - 6*087 h 2 + 1-0342A 4 - 0*05906 ¥. 
Z F - 166*73 - 3*827 h 2 + 0*9941 ¥ - 0*05164 ¥. 
From this follows the general expression for the distance of union 
of the system : — 
(1) Z = A, + A 2 (A' - X) + A 3 O' 4 - X 4 ) 
- [Bj -)- B 2 (X' — \) -)- B 3 (X' 4 — X 4 )] . h? 
+ [C 1 + C 2 (V-A) + C 3 (X'*-A 1 )] . 
- [D, + D 2 (A 1 - X) + D 3 (V* - A 4 )] . ¥ 
= A — B . ?i 2 + C . It 1 - DW. 
where X is the reciprocal of the wave-length in micra, X' = X 0 55 , 
and 
A l = 169*16 
B 1 = 4*779 
C x = 0*9593 
D x = 0*05218 
A 2 = 45*237 
B 2 = 5*166 
C 2 = 0*9774 
D 2 = 0*06117 
A 3 = - 1*2 
B 3 = - 0*0395 
C 3 = - 0*03854 
D 3 = - 0*002018 
From this formula the distance of union of the rays C and F is 
calculated for h = 0*1, 0*3, 0*5, and so on. 
h 
z c 
Z D 
Z 0-55 
Z E 
Z F 
0*1 
175*81 
71*46 
69*11 
67*99 
66*69 
0*3 
75*34 
71*04 
68*74 
67*64 
66*40 
0*5 
74*41 
70*25 
68*03 
66*98 
65*84 
0*7 
73*13 
69*14 
67*04 
66*07 
65*09 
0*9 
71*59 
67*83 
65*90 
65*03 
64*26 
1*1 
69*94 
66*45 
64.72 
63*98 
63*49 
1*3 
68*36 
65*07 
63*57 
63*00 
62*86 
1*5 
66*74 
63*90 
62*67 
62*28 
62*57 
1*7 
6.5*49 
63*02 
62*10 
61*93 
62*73 
1*9 
64*60 
62*52 
61*95 
62*04 
63*44 
2*1 
64*07 
62*44 
62*26 
62*64 
64*75 
2*3 
63*87 
62*73 
63*01 
63*72 
66*67 
2*5 
63*81 
63*23 
64*03 
65*13 
69*04 
In fig. 86 the path for the red, yellowish-green, and clear-blue rays 
is represented with ¥ (not h) as abscissa, and zas ordinate. 
