3 



which the caustic surfaces intersect, when the function v" is positive; this 

 condition is satisfied for ordinary systems, and for the systems produced by 

 crystals with one axis ; 



to/' MR/ ytur' is If 5«Mf' 



14. 



On osculating spheroids, and surfaces of constant action ; the extreme spheroids 

 have their centres upon the caustic surfaces, and their planes of osculation 

 touch the developable pencils ; the centre of the osculating spheroid coincides 

 with the focus of the osculating focal reflector or refractor, when the point of 

 contact and the plane of osculation are the same ; the principal foci are cen- 

 tres of spheroids which have complete contact of the second order with the 

 surfaces of constant action ; they are also foci of osculating focal reflectors or 

 refractors which have contact of the same kind with the last given reflector or 

 refractor ; 



i.V=^^V'; V=^ = C (^P=-^-y ; - 15. 16. 

 -^ .-' ^ V sm. {4^1 —A') J 



On foci by projection and virtual foci ; the planea of extreme projection coincide 

 with the planes of extreme virtual foci ; these planes are perpendicular to each 

 other, and furnish in general a pair of natural coordinates ; the tangent planes 

 to the developable pencils are symmetrically situated with respect to these 

 natural coordinate planes, and coincide with them when the system is rectan- 

 gular ; this latter condition iS' satisfied in ordinary systems, and the osculating 

 foci of such systems coincide with the foci by projection ; 



3j" + ?y" -f ^z" \_ _ (cos, n)' (sin. n)\ 



^ S«3x' -{- lnhj' -f- SySz' ' p ~ p, ' Pi ' 



+ h' 



r =: r, (cos. a)' + r^ (sin. a)'; .. 17, 18. 



Theory of lateral aberrations ; aberration from a principal focus ; elliptic or hyper- 

 bolic lines of uniform condensation ; natural axes of the system ; 



