836 
Proceedings of the Royal Society 
Hence dividing off by 
I . . . Tc m j | &2 C 4 • • • h m | ... | b^2 • • 
X I a 2 b s c A . . . k m || . . . k m | . . . | ■ • 
,l m — 2 I 
hm-i | 
we get 
a i^2 C 3 
I tt 2^3 C 4 * * ’ ^rn I I a i^3 C 4 • • • h m j . . . | Ctf > 2 C 3 • • • ^m -1 | 
1 & 2 C 3 ^4 • • ■ C T" 1 | \cgl± . . . l m T" 1 
| «2 6 3 c 4 • • • h m \ P TO | a x 6 3 c 4 . . • k m I P m 
I . . . l m l”- 1 
I «1^2 C 4 •••*«! p m 
, / _ 1 T 1 a 1 ^I c 2^3 • • • lm-1 \ m 1 
I a f ) 2 C 3 * • * ^m-l| f\re 
where in each expression in the right hand side P, w denotes the 
product of the complementary minors of the elements of the last 
column of the determinant in the numerator. 
As before, by interchanges in certain letters, the right-hand side 
changes form, while the left remains unaltered, so that we get m- 2 
additional expressions, which will he seen from the following 
general enunciation : — 
If (123 . . . m- 1) denote any one of the following m - 1 ex- 
pressions, 
| b l c 2 dg . . . l m _i \ m 1 | a l c 2 d s . . . l m _i \ m | a-f 2 d^ • • • lm - i I 
KP W ’ KP m ’ KP m 
then 
1 af.fz . . . h m _ 2 l m _ 1 ' m 1 
KP™ 
- where K = | af) 2 c % . . . k m _-L | , 
(234 . . . m) - (134 . . . m) + (124 . , . m) 
- +(-l) m - 1 . (123. . . m-l) 
. (IV.) 
9. The foregoing results admit of the following interesting geome- 
trical interpretation. 
Let ABC he a triangle the equations to whose sides in ordinary 
co-ordinates are 
a x x -f b-yj + Cj = 0 , a. 2 x + b 2 y + c 2 = 0 , a z x + 6 3 y + c 3 = 0 , 
