58 
MR. A. B. KEMPE OK THE THEORY OF MATHEMATICAL FORM. 
P 
o 
sr 
O 
O' 
Fig. 62. 
331. Where we are considering residuals of collections of r units, if a collection 
of units C is such that every component collection of r units has one and only one 
residual, which residual is also a component of the collection C, then C may be said to 
be satisfied. 
Some Isolated-Triad Systems.—Family No. 1 . 
332. There are certain systems in which every component collection of two units 
has one residual and one only. One such system is met with in the case of the points 
of intersection of a plane cubic with coplanar straight lines; these make isolated 
triads of colli near points such that if a, b, c; d, e, f; g, h, i; a, d, g; b, e, h are 
isolated triads, c,f, i is one also ; i.e., we have (( ab ) (de)) = ((ad) (be)). 
333. In another which I shall now consider the law of distribution of the isolated 
triads is such that if p, q, r, be a non-isolated triad ; and if l be the residual of q, r; m 
of r, p; n of p, q; then l, m, n is an isolated triad ; i.e., we have ((pm)l)=(p(ml)). 
334. It is an immediate consequence of the law of distribution of sec. 333, that if 
s be the residual of l, p (fig. 63), it is also the residual of m, q and n, r. Thus we 
arrive at a collection of 7 units (comprising the isolated triads l, m, n; in, r,p; r, l, q; 
r 
n 
Fig. 64. 
p, q, n; p, l, s; q, m, s; r, n, s), which contains the residual of each of its component 
pairs and is therefore satisfied. The whole collection is shown in fig. 64. 
