ConJlru^iUn of the liecivem, ' 243 



T^he fame other wfe. 



If a different arrangement of the flars flioulc! be felc'6tecL 

 fuch as that in fig. 2. where one ftar is at the vertex of a cone^; 

 three in the cuxu inference of the fuft feclion, at an equal dif- 

 tance from the vertex and from each other •, lix in the circum- 

 ference of the next fe£llon, with one in the axis or center; 

 and fo on, always placing three ftars in a lower iecllon in fuch a 

 manner as to form an equilateral pyramid with one above them: 

 then we fhall have every ftar, which is fufRciently within 

 the cone, furrounded by twelve others at an equal dlflance from 

 the central ftar and from each other. And by the ditFerentlal 

 method, the fum of the two feries equally continued, into 

 which this cone may be refolved, will be 2/2^+ 1 i «^ + |«; 

 where n flands for the number of terms in each feries. To 

 find the angle which a line vx^ paffing from the vertex v over 

 the flars t;, n^ b^ /, &c, to x, at the outhde of the cone, makes 

 with the axis ; we have, by conftrudion, v i in fig. j. 

 reprefenting the planes of the firft and fecond fedlions = 

 2 X cof.30° = (p, to the radius p j, of the firfl fe£lion =; i . Hence 



it will be ^^' — 1 =vf = ivm ; or vm= 2 n/(P^ - i : and, by 

 trigonometry, — ^rr=L=:T. Where T is the tangent of the 



required angle to the radius R (c) ; and putting / =: tangent of 



(c) In finding this angle we have fuppofed the cone to be generated by a 

 revolving reftangvilar triangle of which the line vx^ fig. 2. is the hypotenufe ; 

 but the ftars in the fecond feries will occafion the cone to be contained under a 

 waving furface, wherefore the above fuppofition of the generation of the cone is 

 not ftricftly true ; but then thefe waves are fo inconfiderable, that, for the pre- 

 fent piirpofe, they may fafely be neglefted in this calculation. 



I i 2 half 



