936 
Proceedings of Royal Society of Edinburgh. [sess. 
denoted by n umbrse, and each umbra one of the integers 1 
2 , . . . , m. Thus when n — 3 and m— 2, the array is 
or 
Ill 
112 
121 
122 
211 
212 
221 
222, 
111 
112 
211 
212 
121 
122 
221 
222, 
111 
121 
211 
221 
112 
122 
212 
222, 
according as the first, second, or third umbra is made invariable 
throughout the two rows ; and if p = 1 we have the ‘ incomplete 9 
hyperdeterminants 
111 
122 
+ 
112 
121 
211 
222 
212 
221 
111 
212 
112 
211 
121 
222 
+ 
122 
221 
111 
221 
+ 
121 
211 
112 
222 
122 
212 
whereas if p= 2 we have the hyperdeterminants 
111 122 
211 222 
111 212 
121 222 
111 221 
112 222 
112 121 j y 
212 221 J 
112 211 
122 221 
121 211 
122 212 i 
F- 
111 121 
211 221 
| 111 211 
121 221 
111 211 
1112 212 
112 122 
2121222 
112 212 
122 222 
121 221 
122 222 
which being really identical, have their common expression 
designated a * complete J hyperdeterminant. The general form of 
H p is not specified. Further details would here be out of place : 
the one important point to be noted is the relation between 
hyperdeterminants and the functions of Cayley’s paper of the year 
1843, and this is shortly indicated by saying that the latter 
functions are hyperdeterminants in which y — 1 and n is even. 
