26 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
149. If we desire in the case of a multiple pair associate to indicate the dis¬ 
tinguished pairs, we may employ lines of various sorts in lieu of links, thus :—- 
150. Many of our conceptions and definitions of systems of units involve the idea 
of associates, and the graphical definition of systems by means of links is very 
convenient as enabling this to be visually represented. 
151. In many cases units which we might at first sight regard as associates of a 
system S of some definite form, are really associates of a single heap system H 
accompanied by a system F which remains self-correspondent while II goes through 
correspondences characteristic of the form of S (sec. 128). 
Unified Aspects. 
152. The unit arrived at by regarding any aspect of a collection a, b, c, . . . as a 
single unit, is unique with respect to the collection and may be represented by the 
symbol {abc . . . ). 
153. Two aspects when unified are two distinct units, and not one unit. Thus if 
abc . . . and Imn . . . are different aspects (abc . . ) is not (Iran . . . ). 
154. Every aspect is unique with respect to its unified aspect; for if abc . . . 
{abc . . . ) >-< Imn . . . {abc . . . ), then abc . . . >-< Imn . . . and then (sec. 140) 
abc . . . {abc . . . ) >-< Imn . . . {Imn thus we have Imn . . . {cibc . . .) >-< 
Imn . . . {Imn . . . ); and (Imn . . . ) is not {abc . . . ) (sec. 153), thus {Imn . . . ) is not 
unique with respect to Imn . . ., which is contrary to sec. 152 ; therefore, &c. 
155. If X and p be two unified aspects of the same collection they are unique with 
respect to each other. For, by sec. 152, X is unique with respect to the collection, 
and, by sec. 154, the aspect of the collection which is p when unified is unique with 
respect to p, thus, by sec. 138, X is unique with respect to p. Similarly p is unique 
with respect to X. 
156. The pairs which a unified aspect {abc . . .) makes with a, b, c, . . . respectively 
are all distinguished from each other. For if {abc . . .)a >-< {abc . . .)b, then 
there are units l, m, . . . such that {abc . . .)cibc . . . >-< {abc . . . )blm . . . , 
i.e., such that abc . . >-< blm . . . , i.e., such that {abc . . . )abc . . . >-< 
{blm . . . ) blm . . . , i.e., such that {abc . . . )blni . . . >-< {blm . . . )blm, 
i.e., such that {blm . . . ) is not unique with respect to blm . . . , contrary to sec. 152, 
therefore, &c. 
