ME. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
55 
«! C0 2 
b 2 
a x c 2 , &c. 
b x a 2 
&i b 2 
b x c 2 , &c. 
Ci a. z 
Cj b. 2 
Ci Cg, &c. 
&c. 
&c. 
&c. 
If it be understood that we are throughout dealing with pairs connecting S L S 2 we 
may write any aspect p 1 q 2 thus pq, the position of a letter in the pair taking 
the place of the unit represented by the subscript number. We then have under 
consideration the aspects 
aa 
ab 
ac, &c." 
ba 
bb 
be, &c. 
L . . 
..(X) 
ca 
cb 
cc, &c. 
, 9 , . 
&c. 
&c. 
&c. 
These will be n in number, where n is the number of units in each of the systems Sj 
S 2 . The aspects aa, bb, cc, . . . are undistinguished from each other, but are distin¬ 
guished from all the other aspects ab, ac, be ... . which again are all undistinguished 
from each other. 4 ' 
314. We may regard the aspects as unified, and shall then get a double system P 
composed of the system P x of n unified aspects (act), (bb), (cc) ., and the 
system P 2 of n^-n unified aspects (ab), (ac), (be) . . . . t 
315. The mode of arranging the aspects adopted at X in sec. 313 suggests a mode of 
clothing the units of P which is very convenient for purposes of description, and 
for emphasizing the peculiarities of P; viz., that of regarding the units as arranged 
in rows and columns, the order of which is to be disregarded, so that the rows as 
units form a single heap system, as also the columns. 
316. In fig. 60 we have a system P of the species considered, in which n = 4 ; the 
asterisks represent units of P 1? the dots those of P 2 . 
# 
• # 
# 
Fig. 60. 
317. The pairs of P form a system of multiplicity 13 ; viz., we have the 13 sorts of 
pairs given by the following diagrams, where in the last seven cases the top unit in the 
first column, and the bottom in the second are those composing the pair considered. 
(0 * • . (2) • * , (3) • • , (4) *, (5) ; , ( 6 ) : , 
<0 * * . (8) * :, (9) : * , ( 10 ) ;■* , ( 11 ) ; * , ( 12 ) ; ; , (is) ; ; . 
* If we confine ourselves to the latter system of aspects, we might regard them as aspects of either of 
the systems Si and S 2 ; but the aspects an, bb, cc . . . . must be regarded as connecting pairs of the two 
systems Si and S 2 . 
f We should obtain a system of the same form as P if we took Si, and the unified pairs of Si. Here S t 
takes the place of P b and the unified pairs of S take the place of P 2 . 
