312 ilfr J. Larmor, On Graphical Methods [May 27, 



A construction of this kind enables us with great facility to 

 trace the course of the refraction as the incident focal lines 

 gradually alter their positions. For example, we see that there is 

 always one and only one position of the focus for an incident 

 stigmatic pencil, along a given axis, such that the refracted pencil 

 is also stigmatic ; a proposition which is obviously of importance 

 in a general theory of photographic combinations of lenses. 



9. General Problem. When the thickness of the optical 

 system across which the filament of light passes is not negligible, 

 the graphical treatment is not so simple. It has however been 

 shown ^ that the optical effect of such a system may be precisely 

 imitated by the effect on a straight filament of two definite thin 

 astigmatic lenses mounted on the same axis at a definite distance 

 apart ; and the relations of this simple system can be represented 

 in a graphical manner by aid of the constructions above. It will 

 be more convenient however to combine deviation with the 

 convergence produced by the lenses, so that the axes of incident 

 and emergent pencils may not be identical. The constants of 

 the filament after refraction through the first thin lens may now 

 be graphically constructed by aid of three fixed points; then they 

 must be transformed to a new origin at the second lens ; and finally 

 by another linear construction the constants of the emergent 

 filament may be obtained. 



The complexity inherent in the treatment of optical systems of 

 sensible thickness arises solely from the trouble involved in trans- 

 ferring the analytical constants of the filament of light from one 

 point to another of its axis as origin. This operation can be done 

 graphically just as well as analytically, or better : but the process is 

 necessarily clumsy. Probably the best available construction for 

 this purpose is the one indicated by Maxwell I 



[10. The development of the construction by Pascal's hexagram 

 in § 2 places the theory in a very simple light. The problem under 

 consideration is that of the conjugate focus in two-dimensional or 

 cylindrical optical systems ; or of the conjugate primary focal line 

 in systems which have a plane of symmetry, or which possess the 

 (for this purpose) equivalent property that the focal lines of all 

 pencils which are stigmatic at incidence have common directions. 

 We are supposed to know from observation the positions of three 

 pairs of conjugate points, on the straight axes of the pencil in the 

 uniform media at the two ends of the optical system. Let the 

 three incident foci be marked off on any line at their proper 

 distances apart, say A^, B^, Cj; and let A^, B^, G^ represent their 

 conjugates marked off at their proper mutual distances on any 



1 Proc. Land. Math. Soc, xxiii. 1892, p. 172. 

 ^ loc. cit. 



