ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 281 



the medium in which the objects are, /A, that of the medium in which the 

 images are, then 



x b^ _ a.,0, x 



' 



approximately, when the objects are small. 



Since a, is the image of a lt the reduced path of the ray a&a* will be 

 equal to that of a^a,, and the reduced paths of the rays o^o, and aj^a., will 

 be equal. 



Also because b^ and bfy are conjugate foci, the reduced paths of the 

 rays 6,0,6, and b^b^, and of /JjOj/J, and &o.j8, will be equal. So that the 

 reduced paths 



6,0, + o,6 2 = 6,a, + a.h, 



. . a,6, + ajS, + 0,6, + a^ = aj8i + Oj&j + a a & 2 + a.j/8,, 



these beuig still the reduced patfis of the rays, that is, the length of each 

 ray multiplied by the index of refraction of the medium. 



If the figure is symmetrical about the axis, we may write the equation 



/*i ( A - A) = /* (j/8i ~ 0*6,), 

 where aj8,, &c. are now the actual lengths of the rays so named. 



Now a& = A& + i (a^ + 6g8,) f , 



so that o^ ^7~A x 



x 



and 



, , , a,o, x 6.J/8., 



Simikrly ^ 3 (a A - A) = / 



,. a,a, x 



So that the equation u,- 



VOL, I. 36 



