486 
ASTRONOMY: H. RAYMOND 
values of [pp] are the three roots of the cubic 
(^-X) D E 
D (B-\) F =0 (B) 
E F (C - X) 
and the corresponding values of /, m, n, are derived by inserting these 
roots successively in the set of equations 
(^-X)/+ Dm+ En = 0 
DI+ (B-\)m+ Fn = 0 (C) 
El-\- Fm+{C-\)n = 0 
The coordinates x',y,z^ are referred to the unknown center of gravity 
of the velocity-figure. We can observe only the velocities x, y, z, 
relative to the moving sun. But we readily find that 
Ix'x'] = [xx] - [x] ■ [*] [x'y'] = [xy] - [x] • [y] 
[y'y'] = [yy]-\y]-[y] [x'z'] = [xz] - [x]-[z] (D) 
[z'z'j = - [s] . [z] bV] = M-b]-W 
The quantities [x], [y], [z], are the negatives of the coordinates of solar 
motion, according to the method of Bravais. 
We are limited to the use of the proper-motions alone, or the radial 
velocities alone, since to combine them requires a knowledge of indi- 
vidual parallaxes which we do not possess. The rectangular components 
of the proper-motion part (tangential to the celestial sphere) of fhe 
stellar motions, resolved as before, may be expressed in terms of the 
proper-motions m, m', and the spherical coordinates of the stars, while 
fjL, ju' may be expressed in terms of x, y, z. Designating the tangential 
part by subscript Z, we readily obtain 
Xi = x(l —cos^a cos^S) — y sin a cos a cos^S — z cos a sin a cos 5, 
and similarly for yi and Si. From these x, Xi, etc., are found by multi- 
plication. The means of these quantities are found by integration over 
the sphere. They are, cleared of fractions, 
3[m] = 2[a;] 15 [xiyi] = 7 [xy] 
3[yi] = 2[y] 15 [xiSi] = 7[xz] 
3[z^]=2[z] 15[yi2i] = 7M 
lS[xiXi] = S[xx]+ [yy]+ [zz] ^ 
15[yiyi]= lxx] + S[yy]+ [zz] 
15[sisi]= [xx]+ [yy] + S[zz] 
The admissibility of these integrations depends upon certain con- 
ditions, which may be stated in the form of three hypotheses: 
