51 



pencils are not always perpendicular to each other, and therefore are 

 not always fit to be taken for rectangular coordinate planes, however 

 remarkable they may be in other respects ; but the planes of extreme 

 projection, determined in the present number, possess this important 

 property, and may be considered as furnishing for any given straight 

 ray of an optical system, ordinary or extraordinary, (except the prin- 

 cipal rays,) two natural coordinate planes, which contain the given 

 ray, and are perpendicular to each other. And whenever the deve- 

 lopable pencils are also perpendicular to each other, the tangent 

 planes to these pencils will coincide with the planes of extreme pro- 

 jection, and the extreme foci by projection will be contained upon 

 the caustic surfaces. This perpendicularity of the developable pen- 

 cils requires that there should exist a series of surfaces perpendicular 

 to the rays of the system, and having for their differential equation 



aix -f H'hy + yJ2 = ; ( W") 



and reciprocally when this equation is integrable, the perpendicu- 

 larity here spoken of, exists, and we shall say that the system is rec- 

 tangular. This condition is satisfied in the case of ordinary systems, 

 because, for such systems, the differential equation {U") of the sur- 

 faces of constant action becomes 



. 03 fi'ii. JF= wi (»&x 4- % + y3z) = 0, . --^ - !■ 



and consequently coincides with the equation {tV'"), m having the 

 same meaning as in the thirteenth number ; the rays of an ordi- 

 nary system are therefore perpendicular to the surfaces for which 

 the function V is constant, and their planes of extreme projection 

 are touched by the developable pencils. We may also remark that 

 for such systems X, = \, and the osculating foci coincide with the 

 foci by projection. .- ,rt., f ;.:.j v •• ,^4 -f. •»% a « 



18. There is yet another kind" of fbcf which we shall call Virtual 

 Foci, and which it may be interesting to consider, because they con- 



H 2 



