Aberrations oj a Symmetrical Optical Instrument. 281 



A#/ = Ayi' — O, as might be expected; since the image- 

 point must now coincide with the normal magnified position 

 given by (x{ , ?//) in the final image plane. 



The expressions for the coordinates of the point in which 

 the ray, after refraction at the zth surface, cuts the image 

 plane, in the general case when the errors are uncorrected, 

 may be easily written down. 



Denoting 



2 © ;> X ©*Ui, 2 ©iW, 2 - (- - —\ 



and XBtaW-ig-JL)} 



by A, B, C, D, E respectively, and putting yi = so that /x 

 also vanishes, we have, if M be the linear magnification, 

 ir/=Ma?i and the ^-coordinate of the required point is 



M«, [l - -^3 (Aft (ft 2 + ^7i 2 ) + B (XhJ (3ft 2 + Vl 2 ) 



+ (3C-D)(XAJ 2 ft+E(Xfc 1 ) 3 )], 



where \ is given by (13). 

 The ^-coordinate is similarly 



M^[l- gJji(A*(ii»+tfl + 2Bf 1 7 h (X/, I ) + (C-D)a*i)'ij,]- 



We shall conclude by deducing as illustrations of formulae 

 (18) and (20) the expressions for the deflexion of the primary 

 and secondary foci from the focal plane in the case of a 

 small parallel pencil incident centrically on a lens of index /x 

 at an angle <f> with the axis. 



Here for the first refraction, since \/h l =tar\(f), 



Ai(— )= ^— g — , the deflexion of the primary focus is given 



Vi/ 2p \ r^fju J J '2fju 2 fir! 



At the second refraction similarly 



or tan 2 0/3 3Q-1) ii~l \ 



