Mr. J. Bridge on the Oblique Aberration of Lenses. 343 



object meets a flat screen after refraction, both object and screen 

 being perpendicular to the axis of the lens, and the points where 

 they intersect the axis being conjugate foci. The formulae give 

 immediately the amount of the diffusion, or spread of the rays of 

 any pencil over the surface of the photographic plate, and the 

 effects due to obliquity and aperture respectively are easily deduced. 

 Though they are adapted directly to the case just mentioned, they 

 may without much difficulty be made to suit any other. The fol- 

 lowing investigation might, in fact, be accurately described as a 

 general view of the theory of spherical mirrors and lenses ; this 

 will appear by the deduction of the most important cases usually 

 treated, as well as that which is the immediate object of inquiry. 

 I cannot find that the method of reference to coordinates in a 

 plane perpendicular to the axis at the prime conjugate has been 

 previously adopted, nor that the problem has been solved in its 

 general form*. 



To find the direction of a ray after refraction at a spherical 

 surface. 



Let X, fi, v be the cosines of the angles made by the incident 

 ray with three rectangular coordinates having the centre of the 

 sphere for origin ; x, y, z the point of incidence of the ray ; m 

 the refractive index. 



Then -, -, — are the direction-cosines of the normal 

 r r r 



\ — [ ^ \ = sin 2 angle of incidence, 



fiz—vy, &c. proportional to the direction-cosines of the plane 



of incidence. 

 Let X', fxl, v* be the direction-cosines of the refracted ray ; 

 then the equations for determining \', fjl, v 1 are 



(1.) X'(fiz — vy) + /jJ{vx—Xz) + v'(Xy—fix) = 0. 



(2.) X'x +/jJy +v'z =r 



. A 1 /, \x + fiy + vz)f\ . 



V 1_ ^V 1 ^ -) = A su PP° se - 



(3.) \ 2 +^' 2 +V* 2 =1. 



These equations express, — that the refracted ray is in the 

 plane of the normal and the incident ray — the proportionality of 

 the sines of the angles of incidence and refraction — and that 

 \', /a', v' are direction-cosines. 



Whence 

 \' (fir 2 — y .\x -\- f^y + vz) = fi' (Xr 2 — x.Xx + fiy + vz) — A(Xy — fix), 



* For previous inquiries into the subject of oblique aberration, see Pot- 

 ter's ' Optics,' and Clairaut, Memoires de Vlnstitut, 1762. 



