1889-90.] 
and similarly 
Dr T. M air on Pascal’s Theorem. 
19 
£34 : ^34 ’ £34 • • l^3 C 4i * \ C , 2 , a ^ ’ 1^3^41 > 
an( ^ £56 ’ VbQ ’ £56 1 : l^5 C 6l * j C 5®ei : ‘ 
Consequently the condition may be written 
Multiplying by 
l«Ah 
l^3 C 4l 
|i! 3 a 4 | 
l«AI 
= 0 
l^5 c e! 
w 
K 6 el 
a i h 
C 1 
a s bo 
C 3 
*4 h 
we obtain 
|a 3 &iC 2 | 
|<x 4 6i<? 2 | 
i a i^3 C 4! i a 1^5 C §| 
KVal 
0, 
and on removing from this the multiplier just used, we have the 
condition in the final form 
= 0. 
KW l»iW 
KVJ KW I 
It is shown to agree with Cayley’s by transforming the determinant 
of the latter into an aggregate Of products of complementary 
minors.* 
* Since the order of the three points is immaterial, we see that 
I | a A C 3l 
|a 3 & 5 c 6 | I 
Msftl \a 5 b 3 c 4 \ I 
I | a 3^4 C ll l a 1^5 c 6l I 
1 | a 1^2 C 4l 
|a 4 & 6 c 6 | 1 | 
\ a A4 KMI 1 1 
1 htol \ a A c e\ 1 
It will be found that the second of the three determinants is got by using 
l a A c 6l instead of | cq& 3 c 4 | for our multiplier and divisor in the foregoing trans- 
formation, and the third by using similarly | a z b x c^. This is one proof of their 
identity. Another consists in pointing out the simple fact, that to establish 
the identity of the first two is the same as to prove that 
K^sl-KVel - KVJ-KVel + KVsi-hMsI - KW-hteM, 
— a well-known identity of Bezout’s, and, curiously enough, Cayley’s second 
lemma in this very paper. 
