514 ASTRONOMICAL PROBLEMS. 
If the star » Ursae Majoris has no “proper motion,” we can 
calculate the R. A. and N. P. D. for any date past or future without 
reference to its annual rate of variation, hence no further observations 
are required to determine its position. If it has a “proper motion” 
the exact amount and direction of this can be ascertained by a com- 
parison of its calculated and its observed positions. 
Let P and P’ be two positions of the Pole c 
of the Heavens, C the centre of 2nd rotation. 
PL 
CH =CQ=90°. Then the are HQ measures N 
the angle PCP’, O being the centre of the 
great sphere. 
LEAP’. QO IPP’, IED 
Then HQ= Diy aD Te 
; for se ee = 2 
. angle PCP’= PW = sin COP} L 
12) 
but annual movement of P=20-09 
and angle COP =29° 25’ 47”. 
log 20:09=11°3029799 (10 added to characteristic) 
—log sin 29° 25’ 47” = 9°6913959 
log HQ (angle PCP’)= 1:6115840, 
whence the angle PCP’=40"886 annual variation of PCP’ for simplicity’s sake 
taken at 40’°9 in all these problems. 
Problem IV. 
Without further information about the star 7 Ursae Majoris beyond 
that known on the 1st January 1887, the calculator now proceeds to 
find the N. P. D. of this star for the 1st January 1895. 
CP 12! BS, 
Variation of angle C=8 x 40"-9=5' 27” -2 , 
.. the angle P'Cy= 77" 18’ 207-2 on 1. 1. 95. 
log cos P’Cn= 9°3419302 Cn =36° 31’ 6” 
log tan P'C= 9°7513982 Istarc= 7 415 
log tan Ist arc=19-0933284 2nd arc=29° 27’ 45 
log cos 2nd arc= 9:9399062 
log cos CP’= 99399977 
19°8799039 
—log cos lst arc= 9°9966880 
log cos P'7= 9°8832159. 
By calculation 2. Py =405 91 457-87 
given in Naut. Alm. 40° 9’ 45°61 
IDNR OF © O28, 
Similarly we can calculate the N. P. D. for the same star for the 
Ist January 1755, 132 years before 1887. 
132 x 40”-9=1° 29’ 59”. 
-. angle P'Cy="7° 12/ 53”— 1° 29/59" 
=75° 42’ 54”. 
