of the 'Eye-pieces of Telescopes. 5 



Prop. I. To find the proportion of the tangents of the angles 

 made by the axis of a pencil with the axis of a lens before and 

 after refraction. 



Let AF, FG, GE, Fig. l, be the course of the ray, and let FG' 

 produced meet the axis in D : draw FH, GK, perpendiculars to the axis : 

 then our object is to compare the tangents of the angles FAH, GEK ; or 



GK AH 



to find the value of -vr^ . ^ . Let FH = a, BC = t, AB = b, CE = c, 



BD = x; and let B, C, X, be the values of the latter quantities when a = o. 

 Draw Fh perpendicular to AG produced. Then 



f^u fi a.HK t-BH-KC , 

 Cr A = a — IxLi = a JTjT' = « — « • y ' 



omitting small quantities in the estimation of HD ; 



GK _ t_ BH+KC 



Now I will remark, that we may at once omit t in this expression, though 

 it is larger than the next term which is to be preserved. The reason is 



that -y is independent of the aperture, and as it is our object to examine, 



among the small parts, only those which depend on the aperture, there 

 is no necessity for preserving a small constant term. The product of t 

 and small variable quantities being extremely small, is, of course, to be 

 neglected. It appears, then, that though in the beginning of this and 

 other investigations, we shall be obliged to take t into account, we may 

 always expunge it in a subsequent stage. When multiplied by a constant, 

 the product is to be rejected because small and independent of the aperture: 

 when multiplied by a variable, which is necessarily small, the product will 

 be of an order smaller than the quantities taken into account. Hence, 



GK BH+KC 



FH ~ ^ '^ X ' 



