156 Proceedings of Royal Society of Edinburgh. [sess. 
A = 0 we also get two systems, viz., the tangential coordinates of 
the two lines represented by (5) when A =0. It is thus seen that 
the squares of a, b , c, A are each of them a factor of the eliminant. 
13. Muir has obtained the eliminant of the system (7) in several 
different forms. To these may be added the following : — 
A/QR + 9 x/KP + hjvq = g -fgh 
g hca \ gab' 2ggh+faa'b'c 
Jibe' g fab' 2ghf + gbb'c'a 
gbd fca \ g 2gfg + hcc'ab 
fgh 2fgh + g 
= 0 
( g 2 + 2 gfgh - aa'b'cf 2 - bb’c'ag 2 - cc'a'bh 2 ) 2 
= i{g 2 - abca'b' c)(f 2 g 2 h 2 + 2 gfgh - g 2 h 2 bd - h 2 f 2 ca -f 2 g 2 ab') 
where 
and 
2g = abc + a'b'd 
P = be' -f 2 , Q = ca - g 2 } ~R = ab' - h 2 . 
14. The first of these forms was originally found by applying to 
(7) the method used in § 8. But the first and second forms are 
most readily found by transforming (7) to the form 
cos a = 
x 
Jbc n 
cos j3 = 
h 
\J ab 
and then eliminating a, /?, y by means of the obvious relation 
cos (a + p + y) = 
Jabc a'b'c * 
The third form may he derived from the second by multiplying 
columns 1, 2, 3 by faab'c i gbb'c'a , hcca'b, and subtracting from 
g times the fourth column, and is interesting as showing not only 
how the eliminant of (7) degenerates to a 2 b 2 c 2 A 2 when a', b\ c = 
a, b, c, hut also that the eliminant of (7) is a perfect square when 
either a'b'c' = abc or 
1 
a 
T 
a' 
9 
1 
