Chap. 9.] Aberration. 261 



pie proportion, but the triplicate of the femi-aper- 

 ture*. Thus, if P V and p V reprefent the fame 

 glafs with different apertures P H, p E 3 the latitudi- 

 nal 



* Let K a, L b (Plate XXV. fig. 59.) be two parallel rays very 

 near to each other, incident on the concave refracting furface 

 A C B at a, b, it is required to find m, the interfection of the rays 

 after refraction. Draw E I, E H, the fines of incidence and re- 

 fraction ; and from the point H draw H G perpendicular to the 

 radius E b, and from G draw G m parallel to the incident rays, 

 and cutting the refracted ray /*in m, m is the point required. 



For A a En a m c + ac m=:a m c-\-c E b+c If E .-. A a E 

 c b E~<z m e-\-c E increment of the angle of refraction, and 

 c E ^rrincrement of the angle of incidence .. increment of the 

 angle of incidence : increment of refraction : : a E b : a E b-^-a m b. 



.. a E b : a m b : : increment of incidence : increment of refrac- 

 tion increment of incidence. 



But when the fines of angles are to each other in a given ratio, 

 the increments of the angles vary as their tangents. 



a E b : a m b : : tangent of incidence : tangent of refraction 

 tangent of incidence. 



But H G is the tangent of refraction H G, and OG is the 

 tangent of incidence O b G. 



.-. a E b : a m b : : O G : H O. 



With m as center, and m b radius, defcribe the arc b d, then 



. ab bd 



a. E b : a m b : : - : - 



E6 bm 



Now*J: </::: JH. 



Hence b M : b H : : O G : H O, and G m joining the points 

 G and m is parellel to the incident rays, and confequently by 

 drawing from G a parallel to the incident rays, the point of inter- 

 fedtion of the refracted rays is determined. 



Hence, fince b H b E xcof. of E b H, the radius being unity, 

 the diftance of the interfection from the point of incidence is 

 found, by making as the difference of the tangents of incidence 

 and refraction to the tangent of incidence, fo is radius of the fur- 

 face multiplied into the cof. of refraction, to the di-ftance of the 

 point of interfection from the point of incidence. 



The diftance m of the interfection m g from the axis C F, varies 

 as the cube of the femi-aperture of the fpherical furface. 



S -? For 



