77 



General series and approximate formulae for the aberrations, lateral and longitudinal; princi- 

 pal foci, and axes of a plane system . . . 109,110,111. 



Properties of the rectangular trajectories . ■ ■- '^r _ 2j2^ 



Least linear space, into which can be collected a given parcel of rays near an axis of a plane 

 system . . . . . . !- 113- 



On finding the system by means of the caustic . . ' i]4>. 



On plane emanating systems; general theorem respecting the focal lengths of plane reflecting 

 and refracting curves, ordinary and extraordinary . . . 115. 



Plane curves having a given caustic ; focal curves . . -. 116. 



XXIV. On developable systems. 



A developable system is a system of the first class, in which the rays have a developable pen- 

 cil for their locus ; equations of a ray ; condition of developability . . 117. 



Formulae for the caustic curve . . . . 118. 



Aberrations ; formula for the radius of curvature of a curve in space ; principal foci and axes 

 of a developable system .. . . . 119,120,121. 



Remarks upon some properties of developable pencils, considered as curve surfaces, 122, 123, 



124. 



XXV. On undevelopable systems. 



Generalisation of the results of the IXth section, respecting the tangent plane, the limiting 

 plane, the virtual focus, and the coefficient of undevelopability . . 125. 



Virtual cauitic, and axes of an undevelopable pencil . . 126. 



Directrix of the pencil ; every undevelopable surface composed of right lines may be gene- 

 rated by one of the indefinite sides of a rectangle of variable breath, whose other indefinite side 

 constantly touches the directrix, while its plane constantly osculates to that curve 127, 128- 



Pencils having a given directrix ; isoplatal surfaces . . 1 29, 1 30. 



The surfaces of centres of curvature of an undevelopable pencil, are the enveloppe of a series 

 of hyperboloids ; formulae for the two radii of curvature ; these two radii are turned in opposite 

 directions; tines of equal and opposite curvature . . . 131 



When the ray is an axis of the undevelopable pencil, the locus of centres of curvature is a 

 common hyperbola ; point of evanescent curvature at which the normal to the pencil is an asymp- 

 tote to the hyperbola of centres ; point in which the ray touches the directrix ; these two points 

 are equally distant from the focus .... 132. 



On every surface whose curvatures are opposite there exist two series of lines, which may be 

 called the lines of inflexion ; properties of these lines; on the surfaces of least area, the lines of 

 inflexion cut at right angles . , ... 133. 



