446 Transactions of the South African Philosophical Society. 



and 288 V 2 being 



a- 



g 2 

 1 



C 2 



1 



h* 



g 2 



or W say, 



he succeeds, by skilful consideration of special cases, in determining 

 the lengthy expression (72 terms) which is the quotient of N by W. 



In 1859 Cayley * drew attention to Sylvester's very curious result, 

 but made no advance in elucidation of it. 



The object of the present paper is to show how the subject may 

 be viewed altogether apart from geometry, and to open the way 

 towards a complete understanding of it. 



2. Taking any vanishing axisymmetric determinant 



(11) (12) 



(12) (22) 



(In) 



(2ra) 



(In) (2n) . 

 and writing its adjugate in the form 



[11] [12] 

 [12] [22] 



[nn) 



[In] 

 [2n] 



or A say, 



[In] [27i] 



[nn] 



we see that since the two-line minors of the latter contain A as a 



factor we have 



[11] [22] = [12]*, 



[11] [33] = [13]*, 



[11] [nn] = [In] 2 . 



To these we may prefix for completeness' sake the truism 

 [11] [11] = [ll] 2 , and thus have 



n/[H] = V[22] . V[§3] 

 [12] [13] 



n/[11] [ii] 



Consequently, since 



A = (11) [11] + (12) [12] + ... + (ln)[l»], 



= (12) [12] + (22) [22] + ... + (2n)[2n], 



[In] 



or 



or 



* Cayley, A., "Note on the Value of certain Determinants, . . ." — Quart. 

 Journ. of Math., iii., pp. 275-277; or Collected Math. Papers, iv., pp. 460-462. 



