THE THEORY OP OPTICAL INSTRUMENTS. 



and the characteristic function from 72, to 72, = 7 (a function of a,, 6,, a,, &,), 

 the reduced path from P, to P, is 



U=n l r l + F+/i,r, (2). 



HIM quantity is stationary with respect to variations of a,, 6,, a,, 6,; therefore, 

 differentiating with respect to these variables, we get 



-.SCS+SE.O u^+^-oi 



r ' ^ r ' A (3). 



If P, and P, are conjugate foci for rays in the plane of xz, the reduced 

 jwth id stationary in passing from one ray to the next by simultaneous variation 

 of a, and <v 



T /I (a.-z.tt tfFl . d'V i 



Hence a, P^ + j : <&*i + -i ? da* = 



L VI r i / iJ ^i^ 



(4). 



[/A I tc ^^ *' j / \ ^* ' I i Ct r 



i_ I ^~ ^^ - - I J" - I fjuti- ^^ (JLCL "^ 



Eliminating efct, and da, from these equations, and putting 6 for the angle 

 between the ray and the axis, we find 



an equation connecting the values of r, and r, for conjugate primary focal lines 

 formed by rays in the plane of xz. 



For rays in a plane perpendicular to that of xz, we obtain the relation 

 of r,' and r,' for the secondary focal lines, by passing from one ray to the next 

 Itv simultaneous variation of 6, and 6,, 



(6), 



whence 



