^518 PHILOSOPHICAL TRANSACTIONS. [aNNO l/Sl. 



let Y express the semidiameter of the aperture of the lens; the angular aberra- 

 tion of the ray falling on the extremity of the lens, or the angle made between 

 this ray, after being refracted through the extremity of the lens, and another ray 

 or line, supposed to be drawTi from the same extremity of the lens, to the geo- 

 metrical focus of rays diverging from the same radiant point, and passing indefi- 

 nitely near the vertex of the lens, expressed in measures of the arc of a circle to 



the radms unity, will be ^-t ., . , h V ^ . — ^ -U^-JL 4. 



•^' (m — w)« X 2m X f' ' (m — n) x 2m x fV "^ 



(m -\- 2w) X Y^ ___ (4n* + 3mn — 3m^) x y^ ^ (2m + 2n) x'y» {3m + 2n) x y' 

 2m X Fr^ (m — n) X 2wi X QF^ m x QFr ' 2»i x Q^F * 



Where r, the radius of the first surface, is exterminated ; and r, the radius of 

 the second surface, is retained: 



Or, exterminating r, the radius of the second surface, and retaining r, the 

 radius of the first surface, the angular aberration is also expressed by 



m* X y' (2ct 4- ") X y' , {m + 2 n) x y? , (3to + n) x Y' 



(»i — hY X 2f' {m — n) X 2f'k 2m x rvC^ ' {m — n) x 2Qr* 



(2w + 2n) X Y^ , (3?« -f 2n) x Y^ 

 m X QFR 2m X Q^F 



As in these theorems, the principal focus is supposed to lie before the glass, 

 as well as the radiant point, to adapt the theorem to other cases, if the lens be 

 of such a form as that its principal focus lies behind the glass, p must be taken 

 negative : likewise, if the rays fall converging on the lens, or the point to which 

 they converge lie behind the glass, a must be taken negative: lastly, if the first 

 surface be convex, r must be taken negative; and if the second surface be con- 

 cave, r must be taken negative; and if, after all these circumstances are allowed 

 for, the value of the theorem comes out positive, the aberration is of such a 

 nature, as to make the focus of the extreme rays fall nearer the lens before it 

 than the geometrical focus, or farther from the lens behind it: but if the value 

 of the theorem comes out negative, the aberration is of such a kind as to make 

 the focus of the extreme rays fall farther from the lens before it than the geome- 

 trical focus. 



With respect to the application o^ this theorem to Mr. DoUond's cx)mbined 

 object glasses, it is evident that if the aberrations of the convex and concave 

 lenses added together (paying due regard to the signs of the theorem) are made 

 equal to nothing, the two lenses will perfectly correct one. another: but as there 

 are two unknown quantities unlimited in the equation, namely, the radius of one 

 surface of each glass (for p and q are given, as well as m and n) there is room for 

 an arbitrary assumption of one of them, at the discretion of the theorist or 

 artist; which being done, there will remain a quadratic equation, whence there 

 will result two values of the radius, which remains unknown, either of which 

 will produce an aberration equal to that of the other lens. 



