MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
25 
. . . >- <ipqr . . . def. . . , therefore abc ... is not unique with respect to d, e,f, . . . 
contrary to the hypothesis; i.e., abc ... is unique with respect to g, h, i, . . . 
139. If a single unit x be unique with respect to a collection a, b, c, . . . we may 
represent x by the symbol {abc . . .) or [abc . . .] &c. The two symbols {abc . . .) and 
[abc . . .] will, where the different sorts of brackets are distinguished from each other, 
represent two units which are distinguished from each other, and each unique with 
respect to a, b, c, . . . ; or we may use the two symbols to represent two undistin¬ 
guished units p, q which are each unique with respect to a, b, c . . . , the difference 
between the brackets indicating that the units p, q are not identical. 
140. If abc . . . >-< pqr . . . then {abc . . .) >-< (pqr . . .), where {abc. . .) 
and (pqr . . . ) may be the same or different units; and {abc . . . ) abc . . .>-< 
( pqr . . . ) pqr ... If abc . . . -:> pqr . . . then {abc. . .) and {pqr . . .) may or 
may not be distinguished units. 
141. If we consider aspects such as P abc . . ., Q abc . . ., P def. . ., &c., where in 
each aspect one only of the units P, Q, &c., appears, we may represent P, Q, &c., 
by brackets of different sorts, and write the units which are unique with respect to 
P, a, b, c, . . ., &c., thus {abc . . . ), [abc . . .], {def. . . ), &c., w r here P is represented 
by the brackets ( ), Q by the brackets [ ]. 
142. Suppose a = (bc), 6= (c/e), e=(yb), then we may represent a by the symbol 
( ( de) c) or ( (cl {fc) ) c), each symbol representing a and at the same time an aspect 
of a collection of which a is a component. We may have such symbols with various 
different sorts of brackets, e.g., if d=[mn\, a will be represented by ( ( [tow] e) c). 
Associates. 
143. If a, b, c, . . . be any collection of units, and if X be another unit, such that 
the pairs \a, \b, Xc, . . . are distinguished from all pairs which X makes with units 
which are not components of the collection, X may be said to be an associate of the 
collection a, b, c, . . . 
144. If the pairs Xa, Xb, Xc, . . . are all undistinguished from each other, X may be 
termed a single pair associate. 
145. If some of the pairs are distinguished from each other, X may be termed a 
multiple pair associate. 
146. In the special case in which all the pairs are distinguished from each other 
a multiple pair associate may be termed a discrete pair associate. 
147. A single pair associate of a collection can only exist if all the units of the 
collection are undistinguished from each other. Also in the case of a multiple pair 
associate of a collection units of the collection which make pairs with the associate 
which are undistinguished must themselves be undistinguished. 
148. An associate of any description may be graphically represented by a graphical 
unit connected by links with each of the units of which it is an associate. 
MDCCCLXXXVI. E 
