Lord Rayleigh's Investigations in Optics* 47 



The condition that there shall be no aberration requires that 

 the quantity within brackets vanish. In order to discuss it 

 further it will be convenient to express u and v in terms of u\ 

 By the original equations ; y being neglected, 



u cu cr u r - 



V cu cs u s 



LLC 



yt! being written as before for — — 



By substitution of these values in (4) the condition becomes 



This condition assumes a simplified form when r=s, i. e. when 



one face is as much convex as the other face is concave ; it is 



satisfied either by 



c' 1 



V-=0, (38) 



u r v J 



or by 



&* +1 >£ + 7=° < 39 > 



In the first case, by (35) u= — cr ; so that if the first face be 

 convex the incident rays must be convergent. In the second 

 case, 



u = — c(|«/ + l)r = — (fM cos </>' + cos <£) r. 



With general values of r and s, (37) may be written 



which determines the values of u' (if any) for which the aber- 

 ration vanishes. The roots of (40) are real ; and therefore 

 aplanaticism is possible if 



^-(V + V-l)(^J) , . (41) 



be positive. Unless both faces are flat, one must be convex 

 and the other concave. The limits within which a suitable 

 value of u' will secure aplanaticism are found by equating (41) 



