50 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
such that each Xp, row contains one cube and one only of the collection. If S be a 
single heap system, the multiplicity of A will be the number of the systems into 
which the collections of cubes thus constituted break up. 
288. We may represent any one of the collections which when unified is a unit 
of A by a square diagram. Take, for example, the case n = 4 ; we have such a 
diagram as the following :—- 
a 
<5 
c 
d 
a 
a 
d 
b 
d 
i 
c 
c 
a 
b 
c 
6 
a 
c 
cl 
d 
d 
3 
a 
a 
Here the square may be supposed to be a face of the cube normal to v , X and /x 
being supposed to be vertical and horizontal respectively. The left hand letters 
indicate whether the adjacent X layers are a K or b K or c K , &c,, respectively, the top 
letters whether the adjacent /x layers are cy or or c v &c., respectively ; and the 
letters contained in the squares, each of which squares is supposed to represent a 
cube, i.e., a unit of E, indicate which v layers the various cubes lie in. 
289. By regarding the pairs connecting the system E and a system of units n, oj, 
&c., which are all distinguished from each other, as unified, we may get a number 
of systems E^, E u , E^, &c., replicas of each other and of E, and we may consider 
systems arrived at by taking one or more component systems of each of the systems 
E^, E^, Ef, &c. Similarly in the case of A. 
290. Now let a be one of the units of A arrived at by regarding as unified a 
collection of units of E of which (ajj^c v ) is one. This collection of units of E which 
when unified is a, contains no other unified triad of the sort e.g., 
so that c v is unique with respect to a, a K , /y; for when c„ is changed to d v , cy and Zy 
remaining unchanged, a is changed to a unit arrived at by regarding a collection of 
units of E containing as unified, which cannot be a. We may, therefore, 
write 
A = 
We may deal with the other triads concerned in arriving at a in the same way, 
and get a collection of n 3 equations, viz.— 
(«Akk) = • 
(««A) = A 
(«« A <y) = . 
= . 
(«&A) = • &c - 
(«A) = • 
&c. 
&c. &c. 
