on the local arrangement of the celestial bodies in space. 311 
feet equality should be admitted ; and as the equal action of 
the instruments was now ascertained, I calculated the diame- 
ters of several apertures to be given to one of the telescopes 
as a standard, so that the other, called the equalising tele- 
scope, might be employed, with all its aperture unconfined, to 
examine a variety of stars, till one of them was found whose 
light was equal to that of the star to which the standard tele- 
scope was directed.* 
In order to be sufficiently accurate in the calculation of the 
diameter of the limiting apertures, I thought it necessary to 
take into consideration not only the obstruction of incident 
light occasioned by the interposition of the small mirror, but 
also of the arm to which it is fastened, and proceeded as 
follows : 
If A be the diameter of the large mirror; b that of the 
small one; the length of the arm; t its thickness; 
it the circumference, diameter being unity; x an assumed 
quantity for finding the correction; A' the aperture corrected 
for the interposition of the arm ; L the light of the equalising 
telescope ; p the proportion of the light required for the 
standard telescope; D the diameter of an aperture to give 
that light ; D' the diameter corrected for the interposition of 
the arm. 
Then will the diameters of the limiting apertures be had 
by the following equations. xr = ff Ai; ^f^xt==2X; 
A — 2x = A'; A 1 - — 6 S = L; pL~ D 2 — 6®; V/>L+> = D; 
xt = o.y ; D 2j — IT the required diameter. 
* I preferred the limitation of the light by circular apertures to the method of ob- 
taining it by the approach or recess of two opposite rectangular plates, in order to 
avoid the inflections whjch take place in the angles. 
MDCCCXVIII. S S 
