MB. A. B. KBMPE ON THE THEORY OF MATHEMATICAL FORM. 
59 
335. In a system of this species, if we add a unit to a satisfied collection C of m 
units, we must also add m others, getting 2m-j-l in all, before we can get a satisfied 
collection. This may be thus shown :—Let a, b, c, cl, ... . be the units of C, and 
let a unit u be added to C, and let a, b ', c , .be the residuals of u, a; u, b; 
u, c; ... . respectively; then a , b', c',... must all be different units; for if a and 
b' were identical we should have both u, a , a, and u, a', b, isolated triads. If then a 
unit u he added to C, m others a, b', c', cl', . . . must also he added, making 2 1 
in all. This collection of units will be satisfied, so that we need not add any more 
units. To prove this we have only to show that the residual of b', c', any two added 
units is a unit of the enlarged collection. Let the residual of b, c, which will be a 
unit of C be a, then we have the non-satisfied triad u, b, c, and the residuals of its 
pairs are b', c', and a which by sec. 333 form a satisfied, or isolated triad, for they 
are the same thing, so that the residual of b', c , is a. 
336. Now every non-satisfied triad is a component of a satisfied collection of seven 
units; and by the preceding section if there be another unit added we get a collection 
of 15 units, if again another be added one of 31 units, and generally a satisfied 
collection consists of 2” —1 units. Every system of the species considered is of course 
satisfied, and therefore contains 2 m —1 units. 
337. If to any component collection C of one of these systems we add such residuals 
of the different component pairs as are not already in C, and repeat the process on 
the enlarged collection we get a definite satisfied component. The component II 
which consists of all the added units may be termed the complement of C. The 
complement of any pair is their residual. If C be itself satisfied there will be no 
complement. It does not follow if R be the complement of C, that C is the comple¬ 
ment of R, though the complement of R must be a component of C. Thus the 
complement of p, q, n, s, l, in fig. 64 is m, r, but the complement of m, r is p. 
338. Consider a system S of the family containing 2"—1 units. Take any pair of 
S, add to it any unit of S not the residual, add to the resulting triad any unit of S 
not in its complement, add to the resulting tetrad any unit of S not in its complement, 
and so on. Proceeding in this way we get an n- ad, one of several, such that its 
complement comprises all the remaining units of S. The components of three, four, 
&c., units of the n- ad may be termed principal triads, tetrads, &c., of S. A principal 
mad has the same self-correspondences as a single heap of n units. 
339. Let a, b, c, d . . . . k, l be a principal component of a system S, substitute for 
a, b them residual y, for y and c them residual 8, for 8 and d their residual e, and so 
on until we get the pair A, l, for which substitute their residual p, which will be 
unique with respect to a, b, c, d . . . . k, l. We should arrive at p if we took the 
units of the component in any other order. To show this it is only necessary to 
prove it for an interchange of any two units, for by a succession of such interchanges 
we can pass to any fresh order. Let us then interchange any two units c and cl, and 
let £ be the residual of y, d, then we have to show that e is the residual of £ c. This 
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