;,, JOCAL LINES OF A REFRACTED PENCIL. 



but in the opposite direction. Join Z>% and produce to P, where VA 

 meets BD. 



Join CD, and draw OQ perpendicular to CD from 0. 



Make Od, OtT each equal to OQ. Draw dP, d'P cutting OA in b and a. 

 Bisect the angle DOQ by 0. 



Tlu-n Oa*o, 06 = 6, and DOE=<f>, the angle which the first focal line 

 makes with the plane of xz. 



5. If a, b, and <f> are given, the construction is easily reversed, thus : 



Let Oa = fi, 06 = 6, and DOE=<f>. 



Draw aa' perpendicular and equal to aO. Draw ba cutting OD, the per- 

 pendicular to nO from in d. Cut off Od' equal and opposite to Od. I 

 -/' cutting bd in P. 



Draw OQ=Od so that the angle DOE=OEQ. 



Draw CO.Z) perpendicular to 00., cutting Oa in (7 and Od in D. 



Make OZX equal and opposite to OD. Draw DP, Z>'P cutting Oa in />' 

 and - J. 



Then OA = A, OB = B, and OC=C. 



6. If therefore the given data be the radii of curvature of the refra. 

 surface and the angle, <j>, which the plane of incidence makes with the prin- 

 cipal section whose curvature is a, we may determine A, B, C for the refracting 

 surface. 



Then from a, and b u the distances of the focal lines of the incident pencil, 

 and ^, the angle which a, makes with the plane of incidence, we must find 

 A., BI, C t for the incident pencil. 



From these data, by 4, we must determine A t , B lt C t for the refracted 

 pencil, and from these a,, 6,, and <,. 



I have not been able to obtain any simpler construction for the general 

 case of a refracted pencil. 



7. For a pencil after passing through any series of surfaces the construction 

 is necessarily more complex, as ten constants are involved in the general term 

 of the second degree of the characteristic function, which is of the form 



V= ^a^ + Iby? + c l x^ l + iajX,' + &#,' + c^, +px t x t + qxy t + rt/.x, 



