174 THE ROYAL SOCIETY OF CANADA 
any two points L and M on 1, and 7 be the numeric of (LM), then 
since e¢, and & are independent of € and of each other, 
ren = e(rn) = (ae) = (ee: ee = — (ee) = — (ee) 
= a (ee) = (a e)e=7ne 
En = ne. 
(iv) Let €, a, & be the indices 
of three collinear points P, Pj, Ps, 
then will 
P,P. = PP.c + P, Pe 
Draw any straight line 1 co- 
planar with P,P:, and on 1 let fall 
perpendiculars PL, P,L; and P.L», 
meeting 1 in L, ly, In; let 7 be 
the index of 1, and h, M, km the 
numerics of. (PL), (P,L,) (P:Ll:), 

then will 
Li L(+) +LL(4+ fy) = (LiL+ LL.) (2, + ho) 
(LL + LL) = LL, + LL} 
1.6. LiL.) = LL, +L, Lh 
oe P,P.h => PP:/: + P, Ph. 
P,P.en = PP2«, N+P Pen 
(Pi Poe) n= (PPra + P:Pe)n 
Hence 7 being a variable independent of e, « and &, we are, justified in 
postulating 
P, Pye = PP.¢,+P Pe 
(v) Let , m, m2 be the indices of three concurrent coplanar lines 
OP, OP, and OP;, then will 
sin P,OP>2.n=sin POP». -+sin PiOP. 72 
= Let € be the index of any point 
; Rin the plane of P,OP», and h, M, 
i he be the numerics of the perpen- 
dicular distances of R from OP, 
OP; and OP, respectively, and let 
P 7 be the numeric of (OR), then 
will 
sin PiOP, sin POR 
—sin POP,-cos P,OP sim-FOR 
+cos POP, cos PiOP sin POR 
=sin POP, sin P,OP cos POR-+sin POP: cos PiOP sin POR 
—sin POP, sin P,OP cos POR+cos POP; sin P:0P sin POR 
=sin POP; sin P\OR+sin POP sin P:0R 
Oo P, 
