ASTRONOMY: H. RAYMOND 
485 
THE PRINCIPAL AXES OF STELLAR MOTION 
By H. Raymond 
DUDLEY OBSERVATORY. ALBANY. N. Y. 
Communicated by E. B. Frost. May 28, 1917 
Suppose the stars of known motion transferred to a fixed point O, 
and allowed to move with their present velocities for a definite length of 
time. They will have expanded into a cluster, the Velocity-figure.' 
According to the older idea that the stars were moving 'at random/ 
this cluster should be spherical. Since J. C. Kapteyn showed^ the 
figure to be elongated, several methods of finding the amount and 
direction of elongation have been proposed and used. Some of these 
make use only of the directions of observed proper-motions, leaving 
the amounts of those motions wholly out of account, thus avoiding the 
difficulties caused by our ignorance of the distances of most stars; and 
are based on some hypothesis as to the actual shape of the velocity- 
figure. The most notable of these are A. S. Eddington's^ and K. 
Schwarzschild's.^ The method here used makes no assumption as 
to the form of the velocity-figure, and is essentially a process of finding 
its principal momental axes. It thus belongs to the same genus as 
those used by C. V. L. Charlier,^ K. W. Gyllenberg,^ A. S. Eddington 
and W. E. Hartley,^ and H. C. Plummer.^ 
Let the coordinates of a star in the velocity-figure, referred to suitable 
axes through O, — say to the First Point of Aries, to (6h, 0°), and to the 
North Pole, respectively, — be x\ y\ z' \ the direction cosines of an 
arbitrarily chosen line be I, m, n\ and the projection of the star 5 upon 
this line be P. Then the component of motion in the direction /, w, 
is ^ = OP = Ix' + my' -\- nz' . Representing the mean of a quantity 
by enclosing it in square brackets, 
\pp\ = /2 [x'x']-{-m^ [yy]+n' [z'z'] + 2lm [xy] +2 In [x'z'] + 2mn [y'z'] (A) 
The axis of preference is defined as the line for which [pp] is a maximum; 
for the axis of avoidance [pp] is a minimum. 
Newcomb, in his paper 'on the Principal Planes toward which the 
Stars Tend to Crowd,' applies to another problem an essentially similar 
process, and gives a detailed solution by Lagrange's method for a 
maximum or minimum of (A), from which the procedure of this investi- 
gation is adapted. Putting A = [x'x'], B = \y'y\ C = [z'z'\ D = 
Wy'], E = [x'z% F = [y'z^], the maximum, minimax, and minimum 
