(25 
where the »’s represent observed quantities all having the same 
weight, and let the number of equations exceed the number of 
unknowns. 
If now we take arbitrary values for « and y, and substitute these 
in the equations (1), there will remain certain residuals (A). A 
second set of values of « and y will be regarded as more probable 
than the first set, if it gives rise to a smaller value of = A?. 
Therefore, in order to find the most probable values of # and y, 
“the adopted values of these quantities must be made to vary until 
= A’, or, as is commonly said, the sum of the squares of the errors 
of observation 1) reaches its minimum value. 
b). The equations of condition must not be regarded as ordinary 
algebraical equations. 
It is not allowed to 
eliminate unknowns 
from them, to multiply 
some of them by a 
constant factor, etc. 
Let S be an arbi- 
trarily chosen star, 
of which the observed 
proper motion is 
Sie ies 
P the North pole 
of the heavens; 
A (coordinates A and D, at The position of A and the 
. 6 z star S) s . 
distance À from the star S) an = ie ni Rn variable ; they 
arbitrarily adopted position of the ° 
Antapex ; coincide with the most probable 
A aie Pat i bitrarily adopted position of the Antapex and the 
v most probable value of the parall- 
value of the parallactie motion ; factie motion, if certain minimum- 
conditions are satisfied. 
1) The expression is, of course, not literally correct. The true errors of observa- 
tion, and consequently also the sum of their squares, are constant quantities, which 
ean have neither maximum nor minimum, Exectly in the same way the expression 
Er? == minimum, is not literally correct when ¢ is defined as the projection of the 
p-m. on the line perpendicular to the direction of the Antapex. The true meaning 
of the first expression is explained above. That of the latter is entirely analogous. 
