ConJlruBlon of the Heavens* 241 



PROBLEM. 



Thejiars being fuppofed to he nearly equally fcaitercd^ and their 

 number, in a field oj view of a known angular diameter ^ being 

 given, to determine the length of the vfual ^ay. 



Here, the arrangement of the flars not being fixed upon, we 

 muft endeavour to find which way they may be pLiced io as to 

 fill a given fpace moft equally. Suppofe a redangular cone 

 cut Into fruftula by many equidiftant planes perpendicular to 

 the axis ; then, if one fliar be placed at the vertex, and another 

 in the axis at the firft interfedion, 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 fiiall have every ftar within the cone furrounded by eight 

 others, at an equal diflance from that ftar taken as a center. 

 Fig. I. (tab. VIII.) contains four fedions of fuch a cone diftin- 

 guilhed by alternate (hades, which will be fufficient to explain 

 what fort of arrangement I would point out. 



The feries of the number of flars contained in the fe- 

 veral fe£lions will be 1.7. 19. 37. 61. 91. &c. which 

 continued to n terms, the fum of it, by the differential method, 



will ht na-\-n , "—— d' •\-n . ^— • —— d'\ &c. : where a is 



2*23 



the firft term d\ d'\ d"\ &c. the ift, 2d, and 3d differences. 

 Then, fince tf=i, ^'rz6, d''-6, d'" — o, the fum of the 

 feries will be «\ Let S be the given number of ftars; i, the 

 diameter of the bafe of the field of view ; and B, the 

 diameter of the bafe of the great re(5l:angular cone ; and, by 



trigonometry, we (hall have Brn ^^^^ 1 field ' -^^^j fince the 

 Vol. LXXV. I i ' field 



