uk = uk :T:r: — • _. 1/ /.. > ^ a • ('^) 



WHLLS: FORMUI.au for MliRIDIAN RAYS 1 83 



There are two special cases which require consideration: 

 (i) when Uk is infinitely great, and (2) when r^ is large (greater 

 than ten times the focal length) . In the first case, since <pk = —6^ 

 when Uk = ^ ^ 



Sill 0^ = — — . (7) 



Tk 



In the second case both Ck and Ck are large. The formula 



usually given for this case, immediately derived from (7), {4), 



and (6), is 



sin ^K cos V2(«K+1+ gg) 



sin ^K+i' cos \/2(«k: + O * 



The angles are computed as usual, and Uk is computed from the 

 formula 



Uk = i-K-i - Ck_, - dK-i- (g) 



The transfer of origin back to the center is made by use of (5) 

 and (6), thus 



Ck+1 ^ i-K+i - Uk + dK- (lo) 



There are apparently two special cases : ( i ) when the incidence 

 is nearly normal, that is, when d-^ is small, and (2) when the 

 refracted ray is nearly parallel to the axis, that is, when ««: 

 is small, but in both cases formulae (/) to (5) may be used. 

 Although Ck and Ck are small when 0^: is small no precision 

 is lost in Ck+i or in of^+i which do not change in order of 

 magnitude. Similarly when o:k+i is small no loss of precision 

 occurs unless the interval between centers, dcK> is large. This 

 is evident from an expression of sin ^^41 which may easily 

 be derived from (4), (5), and (/), namely 



rK+i sin ^K+i = Tk sin 4 + dcK sin a^+i- 



When a^^i is small, ^^+1 is determined by 0^ Q-i^d the r's. 

 The angles 6 and a are both small only when r is large, a case 

 which has just been considered in formula (8). 



The theory of optical instruments is burdened by too much 

 diversity in matters of convention. It seems inadvisable to 

 depart from the time-honored conventions of geometry and 

 trigonometry. Thus distances measured from left to right, 



