Professor Baker, On the Hart circle of a spherical triangle 121 



to the quadric; by ia, i^' , iy' the Cayley separations YZ, ZX, XY] 

 by (A), (B), (C) the Cayley separations of the pairs of planes 

 meeting respectively in OX, OY, OZ; and by (A'), (B'), (C) the 

 Cayley separation of the plane XYZ respectively from the planes 

 YOZ, ZOX, XOY. Each of these separations is ambiguous in sign 

 and by additive multiples of it, unless we enter into further detail. 

 There are however equations by which all of them are deducible 

 from a, P,y; and these equations may be represented, when proper 

 regard is paid to the ambiguities, by 



a' = ITT + ^ — y, ^' = iiT + y — a, y' = iV + a — /3, 



... sinha , , .,. sinh (^ - y) 



tan (A) = — 7~ r , tan (A) = — , ' , q ^—. 



^ ' cosh (e + a) ^ ' cosh (e + ^ + y) 



where e is such that 



2 tanh e = tanh a tanh ^ tanh y — tanh a — tanh ^ — tanh y. 



And these lead to 



(A') = {B) - (C), {B') = (C) - (A), (C) = (A) - (B), 



which may be used to define the Hart section. 



§ 7. In what has preceded we have stated a sufficient condition 

 for the Hart section, namely that AU, BV, CW are concurrent. 

 It can however be proved that this is also a necessary consequence 

 of the existence of the four sections of the quadric all touched by 

 four other planes, provided we exclude certain particular possi- 

 bilities which are easily stated. Precisely, given three arbitrary 

 plane sections of a quadric, no one of which degenerates into two 

 straight lines, so that the equation of the quadric referred to these 

 and the polar plane of their point of intersection is of the form 

 {abcfgh^xyzf = t-^, in order that these with a fourth section (also 

 not two straight lines) should form a set of four sections all touched 

 by four planes, if no relations are assumed to hold among the 

 coefficients a, h, c, /, g, h, it is necessary that the condition in 

 question (that AU, BV, CW are concurrent) should hold. 



In order that the sections by a; = 0, ?/ = 0, z = 0, ^ = of the 

 quadric {abcdfghuviv\xyztf- = should have four common tangent 

 planes, the cones enveloping the quadric along these sections must 

 be concurrent; if A be the four-rowed determinantal discriminant, 

 and ^4, 5, ... the minors therein, it follows that the necessary and 

 sufficient condition for this is that the equation 



{ahcdfghuvivjVA, VB, VC, VDf = A 



should be satisfied for four choices of the signs of VA, VB, VC, vD. 

 It proves to be possible to examine all the ways in which this can 

 happen, and the result is as stated. 



