1906-7.] Dr Muir on Axisymmetric Determinants. 
135 
XVI. — The Theory of Axisymmetric Determinants in the Historical 
Order of its Development up to 1860. By Thomas Muir, LL.D. 
(MS. received December 24, 1906. Read January 21, 1907.) 
Cayley (1841, May). 
[On a theorem in the geometry of position. Cambridge Math. Journ 
ii. pp. 267-271 : or Collected Math. Papers , i. pp. 1-4.] 
A general account has already been given of this interesting paper — 
interesting as regards the subject, and interesting as being the author’s 
first prentice effort. All that remains to be noticed here is what may be 
called Cayley's series of vanishing axisymmetric determinants. These we 
may write in the short form 
( X ) 12 ( X )l3 1 
(^21 * (^23 1 
(#) 3 1 (#) 32 • 1 
( x y) 12 (^)is (*y)i4 1 
{ xi ./) i \ • ( x v ) 23 ( x y ) 24 i 
( x y ) 31 ( x y) 32 • ( x y)s 4 i 
Mu ( x y ) 42 ( x y ) 4 3 • i 
ill l 
(*2/2) 12 
(*2/2>i3 
(*2/ 2 ) 14 
(* 2 / z ) 15 
1 
(*y z ) 21 
. 
(*2/2)2 s 
( X V Z ) 24 
(* 2 / z ) 25 
1 
(*2/2)31 
(*2/ 2 )s2 
(* 2 / 2 ) 34 
(*2/2)35 
1 
( x V z )n 
( X V Z ) 42 
(*y z )i s 
(*2/2)45 
1 
(*2/ 2 ) 51 
(*2/ 2 ) 52 
(*2/2)53 
(*2/2)54 
. 
1 
1 
1 
1 
1 
1 
• 
if we put 
(x) rs for (x r - x s ) 2 , 
(xy) rs for (x r - x s f + (y r - y s ) 2 , 
(xyz) rs for (x r - x s f + (y r - y s ) 2 + (z r - z s f, 
The fact that they are identically equal to zero is established by showing 
that each one is resolvable into two factors, of which one or both vanish ; 
for example, that the first is equal to 
