1910-11.] The Less Common Special Forms of Determinants. 313 
each consisting of z terms : further , the said function when z < m vanishes 
identically. 
To prove the proposition when m = 3 he takes the previous case 
2^ — 2a ~ 2a j/h (x 2 ) 
and multiplies both sides by 2/y 1? thus obtaining 
2a 1 /5 2 y 3 + 2a 1 y 1 /3 2 + 2 < d 1 y 1 a 2 = 2a 1 2/3 1 2y 1 - 2a 1 /3 1 .2y l , 
in which the preyious case enables him to replace 
2a l7 10 2 by '2a 1 y 1 .'2p i - 2a 1 /3 1 y 1 , 
and 
2 ftyi«2 b y 2/?iyi.2 a i - Sa^y-L , 
with the result that 
2 a 1 /l 2 73 = 2a 1 2^ 1 2y 1 - 2a 1 /? 1 .2y 1 - 20 ^. 2 /^ - 2/? 1 y 1 .2a 1 + 22a 1 /? 1 y 1 (^ 3 ) 
as desired. Similarly, on multiplying both sides of this by there is 
obtained on the left 
2 a l^2T3^4 ‘b Wife's 'b + J 
the last three terms of which have only to be replaced by expressions 
warranted from (x 3 ) in order to give the desired equivalent * for 'Ea 1 /3 2 y 3 S 4 . 
It is then pointed out that when z = m the sum Zcq/3 2 “ tantum a 
determinante differt, quod omnes ejus termini sunt positivi,” and that there- 
fore when z < m the sum vanishes. 
The three sections following (iv, v, vi) are occupied, as has been noted 
elsewhere, with the generalisation of Borchardt’s theorem of the previous 
year. 
Cayley, A. (1857, 10/3). 
[Note sur les normals d’une conique. Grelles Journ., Ivi. pp. 182-185 : 
or Collected Math. Papers , iv. pp. 74-77.] 
In dealing with essentially the same geometrical subject as Joachimsthal, 
Cayley gives, in support of part of his demonstration, the identity 
( *, 
Vi z i ) 
Xj V\ % 
*i a V ? V 
2/A 
h x i 
x \Vi 
< x 2 
2/ 2 *2 - x 
X 2 2/2 *2 
= 
+ 
2/ 2 S 2 
Z 2 X 2 
^22/2 
' *3 
Vs h ’ 
*3 2/3 % 
yf * 8 2 
2/3% 
% 
*32/3 
where the first factor on the left is what Cauchy denoted by S 3 (cc 1 ^/ 2 0 3 ). 
* By an oversight three terms of this are left out by Joachimsthal. 
