242 ■^'■^^'' Herschel o?2 the 



field of view of a telefcope is a cone, we fliall have its So- 

 lidity to that of the great cone of ftars, formed by the above 

 conftrudion as the fquare of the diameter of the bafc of the 

 field of view, to the fquare of the diameter of the bafe of the- 

 great cone, the height of both being the fame ; and the ftars in 

 each cone being in the ratio of the lolidlty, as being equally fcat- 



tered (/^), we have«= a^B^S. And the length of the viiual 

 ray = « - i, which was to be determined. 



(I) We ought to remark, that the periphery ?5nd bafe of the cone of the field 

 of view, in g.^ging, would in all probability feldom fall exactly on fuch ftars as 

 would produce a perfed equality of fituation between the liars contained in the 

 fmalJ and the great cone; and that, confequently, the folution of this problem, 

 where we fuppoie the ftars of one cone to be to thofe of the other in the ratio 

 of the folidity on account of their being equally fcattered, wili not be ilricdy 

 true. But it fliould be remembered, that in fmall numbers, where the different 

 terminations of the fields would moft affeft this folution, the ftars in view have 

 always been afcertained from gages that were often repeated, and each of which 

 confiftcd of no lefs than tc-n fields fiiccellively taken, [o that the different deviations 

 at the periphery and bafe of the cone vv^ould certainly compenfate each other 

 fufficiently for the purpofe of this calculation. And that, on the other hand, in 

 higli g^iges, which could not have the advantage of being fo often repeated, thefe 

 deviations would bear a much fmaller proportion to the great number of ftars in 

 a field of view j and therefore, on this account, fuch gages may very juftly be 

 admitted in a folution where pradical truth rather than mathematical precifion 

 is the end we have in view. It is moreover not to be fuppofed that we imagine 

 the ftars to be aftually arranged in this regular manner, and, returning therefore 

 to our general hypothefis of their being equally fcattered, any one field of view 

 promifcuoufly taken may, in this general fenfe, be fuppofed to contain a due 

 proportion of them; fo that the principle on which this folution is founded may 

 therefore be faid to be even more rigoroufiy true than we have occafion to infill 

 iipon in an argument of this kind. 



n^ 



