( 222 ) 
are treated with least squares, remains in perfect harmony with the 
hypothesis mentioned. 
For a better understanding of the question it may perhaps be useful 
to give here in short AIRY’s reasoning. 
Airy resolves the apparent proper motion into two axes at 
right angles to each other, and represents the components by the 
sum of the components of the parallactic motion of the sun, of the 
error of observation, of the error in the precessional constants and 
of the motus peculiaris. 
Let 7 and U be the directions of those axes, M that of the motus 
peculiaris, H that of the Antapex, 7 and wv, the components of the 
proper motion of a star, t and w the components of the errors of 
observation, m the linear motus peculiaris, A the linear motus paral- 
lacticus and y the distance from the sun, then we have, omitting the 
correction of the precessional constants, the equations: 
h m 
To = — cos (H, T) + — cos (M, T) +t 
? 4 
(A) 
h m 
t= ee U) + —cos(M, U) Hu 
3 
If we resolve the parallactic motion of the sun into three directions 
at right angles to each other, and we substitute 
X cos (X, U) + Y cos(¥, U) + 4 eos (4, U) for heos (H, U) 
X cos (X, T) + Y cos(¥, 7) + Zcos(Z, T) for hcos(H, T) 
then each star will give two equations for the determination of X, 
Y and Z. 
As however the relations between the error of observation and the 
motus peculiaris are not known, AIRY proposes two different solutions 
of these equations: 1°. on the supposition that the irregularities of 
proper motion are entirely due to errors of observation; 20, that 
they are entirely due to peculiar motions of the stars. In either solution 
he supposes that the errors of observation or the motus peculiares 
respectively may be considered as chance-errors, and hence he solves 
the equations in both cases so, that either the sum of the squares 
of the errors of observation or the sum of the squares of the motus 
peculiares is a minimum. 
We confine ourselves to the second supposition, and therefore give 
