1910-11.] The Less Common Special Forms of Determinants. 323 
obtain the first result, learning at the same time that it only holds when 
axisymmetry exists ; the second is self-evident ; and the third follows from 
the fact that the aggregate of the terms which are free of 0, being got by 
deleting 10, 20, 30, is expressible as a vanishing determinant. 
(5) MISCELLANEOUS SPECIAL FORMS. 
Cayley, A. (1845). 
[On certain results relating to quaternions. Philos. Magazine, xxvi. 
pp. 141-145 : or Collected Math. Papers, i. pp. 123-126.] 
Assuming that in each term of the development of a determinant the 
elements are arranged in the order of the columns from which they are 
taken, Cayley points out that if the elements be quaternions 
but 
7 r 7 r 
7T 7 r 
— 7777 — 7777 
= 0, 
77 77 
/ / 
77 77 
7777 — 77 77 
* 0 . 
He is thus led to inquire what the non-zero value is in this latter case and 
in other similar cases. Taking 
77 — x + iy +jz + kiv , 
77 ' j= x' + iy' +jz + Jew', 
77" = x + iy" +jz" + kw" , 
he says it is easy to show * that 
77 77 \ 
j k 
77 
77 
77 
- -2 
3 
i 
j 
k 
77 77 | 
y 
z w 
77 
7 7 
77 
X 
y 
z 
w 
, 
, , 
„ 
„ 
„ 
, 
y 
. 
y 
z w 
5 
77 
77 
77 
X 
Z 
w 
rr 
rr 
ff 
X 
y 
z 
w 
* Probably tbe easiest way is to express the determinant as a sum of determinants with 
monomial elements. In the case of the third order the number of such determinants is 64, 
of which 40 vanish, the sum remaining being 
123 + 132 + 213 + 231+312 + 321 
+ 124+142+ .... 
+ 134+143+ 
+ 234 + 243+ .... 
where rst stands for the determinant whose columns are in order, the r th , s th , £ th columns of 
the array 
x iy jz kw 
x iy' jz' kw' 
x" iy" jz" kw" 
