GO 
MR. A. B. KEMPE OR THE THEORY OF MATHEMATICAL FORM. 
follows at once from the law of sec. 333, for y, h, d is a non-isolated triad and y, S, c, 
S, d, e, and d, y, £ are isolated ; thus c, e, £ is isolated and e is the residual of £ c. 
We should arrive at y if we were at any stage to substitute the residual of any two 
units of the reduced component, instead of always taking the last substituted residual 
as one of the two ; e.g ., after substituting 8 for y, c, we might substitute the 
residual of h, 1c for In, h. The unit y is clearly a component of the complement 
of a, b, c, d .... 1c, l. It is obvious that if we started with the component 
a, b, c, d . . . . 1c, /x we should arrive at the unit l, and generally any other unit, of 
a, b, c, d ... . 1c, l, y would be arrived at from the rest. In fact the whole collection 
a, b, c, d . . . . 1c, l, y is isolated, and each unit is thus a residual of the remainder, 
the only one that there is. 
340. The process of the preceding section may be applied to any component 
whatever of S, with this modification in the case of non-principal components, viz., 
that where the residual of any pair is included in the component so that we cannot 
add it to the latter in substitution for the pair, in such case the whole triad must be 
removed from the component. This amounts to removing any isolated r-ad which is 
contained in the component. If in the result one unit remains, it will be the only 
residual; if no unit remains the component is isolated. 
341. We may classify the units of S relatively to a principal component of n units 
thus, viz., we have 
(1) The principal component of . 
(2) The residuals of the pairs of these 
(3) ,, ,, triads 
(4) ,, ,, tetrads 
n units 
I n 
p in number 
[3 n — 3 
| n 
14 [n-4 
(r) 
(n) The residual of the n units. 1 ,, 
Total. 2"—1 units. 
342. If we represent the units of the principal component (1) by the letters a, b,c. 
. ... , we may represent every other unit of S as a residual of two or more of these 
in the manner indicated in sec. 329. The order of the letters in the brackets will be 
immaterial. The symbol representing the residual of any two units of S will be 
arrived at by aggregating the letters contained in one or other of the two symbols 
