28 © Mr. herschel on the proper Motion 
with the equator, will be the moft northern part of the femi- 
circle mnc. From what has been faid (p. 261. of the paper on 
the Motion, &c.) it follows, that the apparent motion of any 
ftar iS will always be in an arch of a great circle AiSc drawn 
through the apex A and ftar jS : therefore, if the ftar be lei s 
than 90 degrees of the generating circle diifant from its inter- 
feclion with the equator (having more northern declination 
than the apex) as at i, its northern declination will increafe, 
and it will alfo fall within the nodated part of the conchoid 
Ax? y ; but when its diltance from the interfe&ion m is more 
than 90 degrees, as at S, the motion will be towards the fouth, 
and the ftar will be lituated without the nodated part of the 
curve. That the fkr s will fall within the nodated part ap- 
pears becaufe ms being lefs than mn by iuppofition, if m be 
drawn towards E, to defcribe the conchoid the angle A/.vQjvill 
decreafe, and therefore the defcribing point n will be deprefted 
below s as it approaches A. For the fame reafon S will fall 
without; fmce, by drawing m towards Q^, the angle A;«Q 
will become greater than SwQ, and the defcribing point n will 
pals above the ftar S. The application of this theory is very 
fimple ; for inftance, let it be required to find whether any 
given ftar will fall within or without the conchoid. Then, in 
fig. 6, there will be given Pi, the polar diftance of the ftar; 
and QPe, the difference of right afcenlion between the ftar and 
the apex of the fun’s motion A; alfo, the polar diftance PA, 
and declination c A of the point A. Then, by trigonometry, the 
fides iP, PA, and the included angle being given, we find the 
fide As and angle PAi. Again, the lide c A, and angle 
cA/fl = PAi of the right-angled triangle A cm being given, we 
find the hypothenufe Am; and if Am + As be lefs than 90 
degrees, the ftar falls within the conchoid, otherwife without. 
It 
