194 THE ROYAL SOCIETY OF CANADA 
in which , o=An-+vl—In and r=Xm<+ul—Ilm 
ind B'C’ = — apyvm + bdv n+ Crm: 
2A) 
amnm+blinmn<+clmns 
24 
ind AB" = 
REG — et ee Po PORTER 
2A; 
On forming the product of these three index-values, the determi- 
nant factor will be found to vanish and therefore 
ind B/C’ ind F’G’ ind A’”B”=0 
B/C’, F’G’ and A’B’ are concurrent. 
CorROLLARY 1. By homology A’C’, E’G’ and A”B” are concurrent 
as also are A’B’, E’F’, and A’B’; hence the triangles A’B’C’, E’F’G’ 
are uniaxial with the triangles ABC, EFG. 
CoROLLARY 2. Hence the triangles A’B’C’, E’F’G’ and O’P’Q, 
are concentric and their axial lines coincide with the axial lines of 
ABC, EFG and OPQ. | 
Propositions XV to XX are true for figures on any homoeomeric 
surface, whether spherical, pseudospherical or plane. 
[The remainder of this paper as presented, consisted of four parts:—_ 
Ist. A statement of the changes of interpretation of the symbols 
of magnitude in propositions I to IV necessary to render the proofs 
of propositions XV to XX valid for any homoeomeric surface. 
2nd. An exposition of the geometry of Pascal’s Hexagramma 
{ysticum for any homoeomeric surface, in which Pascal’s, Steiner’s, 
Kirkman’s, Cayley’s and numerous related theorems and their duals 
were deduced immediately from propositions XV to XX; accom- 
panied by tables exhibiting the relations of the various lines and 
points determined by the theorems. | 
3rd. A statement of the chief theorems of the modern geometry 
of the triangle in the notation of trilinear and tripunctual indices. 
4th. The enunciation of the postulates determining the interrela- 
tions of point, line and plane indices in the geometry of homaloid 
space of three dimensions. This part will be developed into a separate 
paper. | 
