1910-11.] The Less Common Special Forms of Determinants. 317 
if we use K to stand for 
/?(y'-8')(a'-/3') 4 y(8' -/?')(«' -y') + W-y)(a-V). 
Accepting the first statement, and knowing that the equality referred 
to is 
( y ~ a)(P ~ S) (y - cl)(P' - S') 
(a - £)(y - 8) (a - /3')(y ~ D ’ 
we readily make the deduction that 
* = (y-a)(i8-S)(a'-/3')(y-8') - (a - j8)(y - S)(y' - <*')(/?' - 8') . 
By accepting the second statement, like conclusions may be drawn ; for then 
the elimination of K from any two of the equations involving it must of 
course lead us back to some form or other of the equation with which we 
started. Thus, multiplying K by a and using the first of the four derived 
equations we obtain by subtraction 
0 = (a/? + yS)(y ~ S')(a' - /3') - (ay 4 /?8)(/3' - 8') (a - y) 4 (aS 4/?y)(/3' - y')(a' - S'), 
whence we deduce in the same manner as before that the expression * on 
the right when changed in sign is equal to T r ; and using any pair of the 
remaining equations we reach either the form of , vE r previously obtained or 
one of the two forms derivable from it by means of the simultaneous cir- 
cular substitutions 
/Ly,S = y,8,/3, 
/*',/, S' = y 
Cayley, A. (1858, February). 
[A fifth memoir on quantics. Philos. Trans. R . Soc. London, cxlviii. 
pp. 429-460 : or Collected Math. Papers, ii. pp. 527-557.] 
The second part (§§ 96-114) of the memoir deals with two or more 
quadrics, and forming part of it is a digression (§§ 105-114) on involution 
Similarly, 
1 
0 
a' 
0a' 
1 
a 
1 
0 
0' 
00' 
1 
0' 
1 
7 
7' 
77' 
1 7-0 
7' 
(7 - 0)7' 
1 
S 
8' 
85' 
1 8-0 
8' 
(5-0)8' 
and so of the others. 
In doing this we learn, too, that 
T + K « = ¥ a=0 , ¥ + K(a - {i)='Y a= p 
* This second form of "V may be got directly from the determinant by expanding in 
terms of the two-line minors formable from the first and third columns, and the minors 
complementary to these. Of course we also have 
¥ = (a'j 3 ' + 7'8')(a - fi)[y - 8) - {ay 1 + P r 8 ')(a - y)(fi - S) + [a S' + fi'y'){a - 8)(j8 - 7) . 
