27 6 
parallax of the fixed stars. 
substitute for r' and sin. (<p' — <p) their values in terms of <p, or 
7 r — o- deduced from the equations already stated, by proper 
eliminations, we find 
P = y . v/ 1 — cos. A 2 , cos. (tt — c) 3 (3) 
whence the maximum effect of parallax on the angle of posi- 
tion of any double star is readily computed ; the expression 
is inconvenient for logarithmic computation, but this inconve- 
nience is obviated by the very obvious substitution 
cos. X. cos. (tt — cr) = cos. M ; P = y . sin. M. (4 
We also have 
2 a = $. 
P ^ sin. P 
sin. M " sin. M 
( 5 ) 
from which we may deduce the value of 2 a the maximum 
of parallax when the total effect ( P ) on the angle of position 
is known. 
If we take P = 30' we have 2 a — ^ 
^ cin M 
( 6 ) 
which expresses the amount of annual parallax in any pro- 
posed star which will be indicated by a periodical variation 
of 30' in the angle of position. The smallness of this amount 
is a criterion (mathematically speaking) of the favourable 
or unfavourable nature of the individual star for researches 
of this kind. 
By these theorems the proper times of year for observa- 
tion, and the amount of difference of parallax, appreciable by 
a variation of 30' in the angle of position, have been computed 
for such stars among those observed by Mr. South and 
myself, in the paper above alluded to, as appear favourable 
to the application of the common wire micrometer to this 
enquiry, in the method now proposed. As the number of 
known double stars increases, others may easily be added to 
the list. The Catalogue of 460 double Stars observed by 
