16 REPORT 1875. 



If «', /3', y' be the primitive trilinear coordinates of a point referred to a triangle 

 ABC, then if sin ««, sin /3, sin y denote the spherical coordinates of the projected 

 point referred to a spherical triangle, constituted by planes through the centre 

 and the sides of ABC, 



sin X : sin /3 : sin y : : sin BC : sin CA : sin AB 

 :: ap^a! : hp-.d' : cp^y', 



where p^, p.^, 2^ fire the perpendiculars on the opposite faces of the tetrahedron 

 OABC. 



If the chords of the arcs of the spherical triangle constitute the sides of the tri- 

 angle of reference, 



, /,, . « • f> . „ c . 



et' : /3' : y : : cos j^ sin « : cos ^r sin /3 : cos -^ sm y. 

 Ji Ji Jt 



By this process it was shown that the shadow of a circular cubic has the shadow 

 of the line at infinity for a cyclic arc. The shadow of a Cartesian, which has cusps 

 at the circular points at infinity, has two (and may have three) coincident cycUc 

 arcs in the shadow of the line at infinity. 



5. Formulre were given to determine the tangential equation of the shadow on a 

 sphere of a plane curve, which is itself expressed by tangential coordinates : 



p' : q' : r' ; : OA smp : OB sin q : OB sin?- 



: : p^ sin a amp : p.^ sin 6 sin j : ^3 sin c sin r. 



In the particular case where OA, OB, OC are equal, the formuloe of transformation 

 are identical in form. 



These formulae were applied to deduce equations to spherical curves, and in par- 

 ticular to investigate the projections of the circular points at infinity, and their 

 properties. 



G. This outline of the doctrine of projection on the sphere may be regarded as 

 a separate chapter in spherical analytical geometry, and may suggest further deve- 

 lopments of the subject by following the lead of the great French geometer. 



II. On Cubic SrHEEiCAL Cubves with triple Cyclic Arcs and 



TRIPLE Foci. 



1. On the classi/tcation of cubic cones and spherical ctibics. — There are five cubic 

 cones — simplex, complex, crunodal, acnodal, and cuspidal, to use the nomenclature 

 of Prof Cayley. The singular and non-singular forms have been studied in the 

 canonical and other distinct equations. 



It is here proposed to classify them according to their cyclic planes or arcs — (1) 

 with three single cyclic arcs, (2^ one double and another single, and (3) with a 

 triple cyclic arc. The classification of Newton and PlUcker for plane cubics will 

 thus be imitated ; but the number of groups is much less, viz. three in all. 



As there are three real foci in a plane curve of the third class, it is inferred that 

 there are three real foci in a spherical cubic of the same class, since the tangential 

 equations to both are identical in form ; hence, by reciprocation, there are three 

 cyclic planes in a cubic cone of the third order. The three groups may be con- 

 veniently studied in trilinear equations : — 



(1) Ka^iy = la-\-m^-\-nv, 



(2) Ka'fi =y, 



(3) Ka' = ^, 



where ac is the variable perimeter in each group. The left-hand side defines the 

 cyclic arcs, the right-hand the satellite arc. The symbols denote the sines of the 

 respective arcs. If the variables be interpreted as tangential coordinates, these 

 three groups represent all cubics of the third class both plane and spherical — viz. 

 (1) with three single foci, (2) with one double and one single focus, (3) with one 

 triple focus, 



