VOL. LXXV.] PHILOSOPHICAL TRANSACTIONS. 685 



number of stars; 1, the diameter of the base of the field of view; and b, the 

 diameter of the base of the great rectangular cone ; and, by trigonometry, we 



shall have b = ^ — , ■ ' .. . Now, since the field of view of a telescope is a 

 Tang, n field r 



cone, we shall have its solidity to that of the great cone of stars, formed by the 

 above construction, as the square of the diameter of the base of the field of 

 view, to the square of the diameter of the base of the great cone, the height of 

 both being the same ; and the stars in each cone being in the ratio of the solidity, 

 as being equally scattered, we have n = £/b 2 s. And the length of the visual 

 ray = n — 1, which was to be determined. 



The same otherwise. — If a different arrangement of the stars should be se- 

 lected, such as that in fig. 7, where 1 star is at the vertex of a cone ; 3 in the 

 circumference of the first section, at an equal distance from the vertex and from 

 each other ; 6 in the circumference of the next section, with 1 in the axis or 

 centre ; and so on, always placing 3 stars in a lower section, in such a manner 

 as to form an equilateral pyramid with 1 above them : then we shall have every 

 star, which is sufficiently within the cone, surrounded by 12 others at an equal 

 distance from the central star and from each other. And by the differential 

 method, the sum of the two series equally continued, into which this cone may 

 be resolved, will be 2n 3 -f- lpi 2 + 4-ra ; where n stands for the number of terms 

 in each series. To find the angle which a line vx, passing from the vertex v 

 over the stars v, n, h, I, &c. to x, at the outside of the cone, makes with the 

 axis; we have, by construction, vs in fig. 8, representing the planes of the 1st 

 and 2d sections = 2 X cos. 30° = <p, to the radius ps, of the first section = l. 

 Hence it will be */ <p— \ = vp = \vm ; or vm = 2 */ <p* — \ : and, by trigo- 



nometry, - .- = t. Where t is the tangent of the required angle to the 



radius R ; and putting t = tangent of half the given field of view, it will be 



- = b, the base of the cone. And — ~ = d, will be an expression for vp 

 in terms of vs, which is the mutual distance of the scattered stars. Then having 



— = n 3 + 4-ra 2 + -kn, we may find n ; whence idn — d, the visual ray will be 

 obtained. The result of this arrangement gives a shorter ray than that of the 

 former ; but since the difference is not so considerable as very materially to 

 affect the conclusions, I shall, on account of the greater convenience, make use 

 of the first. 



We inhabit the planet of a star belonging to a compound nebula of the 3d form. 



I shall now proceed to show that the stupendous sidereal system we inhabit, 



this extensive stratum and its secondary branch, consisting of many millions of 

 stars, is, in all probability, a detached nebula. In order to go upon grounds 



