ConfruBion of the Heavens . 241 
PROBLEM. 
The fiars being fuppofed to be nearly equally fc altered \ and their 
number , in a field of view of a known angular diameter , being 
given , to determine the length of the vifual ray. 
Here, the arrangement of the ftars not being fixed upon, wc 
mu ft endeavour to find which way they may be placed fb as to 
fill a given fpace raoft equally. Suppofe a rectangular cone 
cut into fruftula by many equidiftant planes perpendicular to 
the axis ; then, if one ftar be placed at the vertex, and another 
in the axis at the firft interfe&ion, fix ftars may be fet around it 
fo as to be equally diftant from one another and from the cen- 
tral ftar. Thefe pofitions being carried on in the fame manner, 
we fhall have every ftar within the cone furrounded by eight? 
others, at an equal diftance from that ftar taken as a center. 
Fig. i. (tab. VIII.) contains four fed! ions of fuch a cone diftin- 
guifhed by alternate Ihades, which will be fufficient to explain 
what fort of arrangement I would point out. 
The feries of the number of ftars contained in the fe- 
ver al fe ft ions will be 1.7.19.37.61.91. &c. which 
continued to n terms, the fum of it, by the differential method, 
will be n a -\-n . - — - d' + n . - — - • - — 11 d // , &c. : where a is 
223 
the firft term d\ d'\ d /f/ , &c. the ift, 2d, and 3d differences. 
Then, fince a— 1, d ! — 6, d"—(o , d //f — o, the fum of the 
feries will be n\ Let S be the given number of ftars; 1, the 
diameter of the bafe of the field of view ; and B, the 
diameter of the bafe of the great redtangular cone ; and, by 
trigonometry, we Ihall have B =: Ta ^ d 15^ 9 Now, fince the 
Vol. LXXV, I i field 
