38 TRANSACTIONS OF THE [Nov. 4, 
The above steps having been taken, we can proceed to form 
equations from which to determine by least squares the most 
probable values of the unknown quantities. These are the two 
corrections of the assumed proper motion, the parallax, the 
ratio of the masses, and the true distances of the various stars 
from the fixed point of the apparent orbit. 
Let us introduce the following notation: 
p, x= the assumed proper motion on the arc of a great circle, 
and the position angle of that great circle. 
w = the correction required by the assumed p cos x. 
v= the correction required by the assumed psin x. 
t= date of the plate minus 1872.0. 
x= parallax of the system. 
S;, P;, Sy Py= auxiliary quantities for computing parallax co- 
efficients, and having the same signification as on p. 302 
of Dr. Davis’ Parallax paper. 
6,= the mean of the values of ¢ on all the plates for any 
given star. 
Then if we put, for convenience : 
G = 83 Ps +- Sy Py 
r=6)— 0’, d=o—o, 
The following equation will hold true: 
Gr+tteosp. w+7Tsinp. v-+-p’cos(@—p). m+a4+d=0. 
Such an equation can be formed from each star on each plate, 
and from their solution the most probable values of the un- 
knowns, 7, w, v, m and x, can be determined. From these we 
pass at once to a knowledge of the ratio of masses by means 
of the equation : 
M 1+ m 
} cao 0 
and we get the distance from any given star to the fixed point 
by means of the equation: 
o/ =0,—&. 
The total number of equations will be 702, involving 66 un- 
knowns. But the solution of all these equations by least 
squares will offer no difficulty, because we can first eliminate 
the unknown « from all the equations belonging to any given 
star. The reduced normals from all the stars can then be added 
together, and the final definitive values of 7, w, v, m and « 
computed. It will probably be possible to tell in advance, from 
a preliminary consideration of some of the equations, whether 
the investigation is going to give a satisfactory result. This 
