316 Proceedings of the Royal Society of Edinburgh. [Sess. 
roots is not effected without the use of a set of unessential equations of 
which the determinant is the eliminant. 
The interesting fact is noted that when n = 3 the equation can be 
changed into 
a 12 
a ]3 
^23 
^13 
b\2 
a ll~ X 
«12 
a w 
a i2 
a 22 - x 
a 23 
a i2 
a 22 -x 
a 22> 
a !3 
a 23 
a 33 — X 
«13 
°23 
a 33 — X 
i 0 
'23 
'12 
( 7 ) DETERMINANTS CONNECTED WITH ANHARMONIC RATIOS. 
Cayley, A. (1854, February). 
[On some integral transformations. Quart. Journ. of Math., i. pp. 4-6 : 
or Collected Math. Papers, iii. pp. 1-4.] 
This paper opens with two statements in reference to the determinant 
1 a a aa 
1/3/3' (3(3' ' 
7 y 77 
8 8 88 ' 
or \1> say . 
The first is to the effect that the equation 
^ = 0 
asserts the equality of the anharmonic ratios of a, (3, y, S and a, f3', y , S' : 
and the second that the said equation may also be expressed in the forms * 
Ka = - { yS(y' - S')(a - p) + 80(8’ - /3')(a' - /) + /Jy(/3' - y')(a' - S') J , 
K(a-/}) = (S-/3)(/5-y)(y'-S')(a' -/?'), 
K( a - y) = (/3 - y)(y - 8)(S' - /3')(a' - y') , 
K (a — 8) = <y - S)(8 -/})(/?' -y')(a- 8'), 
* These may be established as follows. By separating the terms of K which involve a 
from those which do not, we see that 
K= - 
1 0 0 ' 00 ' 
1 y y' yy' 
1 8 S' 85' 
a determinant differing from ¥ in the first row only, and consequently on multiplying by 
a and adding we obtain 
Y + Iva = 
1 
a' 
= 
P 
0 '-a f 
&8' 
1 
& 
&' 
7 
y - a! 
yy' 
1 
7 
/ 
7 
yy' 
8 
5' - o' 
88' 
1 
5 
8' 
85' 
