184 wells: kormulaic for meridian rays 



and angles measured by anti-clockwise rotation, arc taken as 

 positive in the notation of this paper. 



The vertex of each surface is taken as the origin for its radius 

 and for the object and image distances in refraction at that 

 surface. The slope angles are measured from the rays in to the 

 optic axis, and from the normal (to the surface at the incidence 

 point) in to the axis, and the angles of incidence and refraction 

 are measured from the rays in to the normal. Hence r and 

 if, u and a, n' and a', and d and d' group in pairs having like 

 sign. As r is measured toward the center of curvature, it is 

 convenient to measure both c and c' toward the center. They 

 are defined by equations (O), which have the advantage of sym- 

 metry as regards the signs of u and c. 



The quantities — , r^, and dj^ are constants for each surface, 



and their logarithms may be computed in a special column, 

 copied on a slip of paper, and used directly for all rays. Sim- 



ilarly the logarithms of the ratios may be separately com- 



puted for each wave-length and used for all the rays of the same 

 color. 



For paraxial rays the angles d and a are small quantities of 

 the first order, so that sin d may be replaced by d, and cos Q by 

 unity. In this case the B's and as may be eliminated from 

 equations (7) to {4), giving the Gauss formula 



l^K ^K Ck , . 



Hk+i Tk - (nK+, - nKJCK 



If one must work alone, the constants may be checked by com- 

 puting the paraxial rays both by (/) to {4), replacing the sines 

 by the angles, and also by (//). Decimal trigonometric tables 

 are most convenient, such as those published by the French 

 government. 



