16 
MR. A. B. KEMPE ON THE THEORY OP MATHEMATICAL FORM. 
83. The system Y was described in sec. 81 as consisting of unified aspects of all the 
component collections of S, i.e., the component pairs, triads, &c. It is, however, 
unnecessary so far as defining the form of S is concerned that this should be so. We 
shall see that it is sufficient that Y should include a unified aspect Y of the whole 
system S, and all other unified aspects of S which are undistinguished from A r 
(secs. 194, 195). 
Letters, their Sorts ancl Positions. 
84. Let us now turn aside to consider certain points with regard to the representa¬ 
tion of units by letters. Any letter admits of being repeated as often as we please ; 
each repetition being one of a number of letters of the same sort. When in common 
parlance we speak of letters a, b, c, d, &c., we are really referring, not to the letters 
themselves, but to their sorts. Each letter (not sort of letter) occupies a different 
position on the paper on which it is written or printed ; as in the case of the 
letters, we have a number of positions of the same sort, each called “ the same 
position ” with reference to another of the number. For example, if we have two 
arrays of the same number of letters, e.g., abode and pqrst, we say that “ b occupies 
the same position in the first array that q does in the second,” meaning that the sort 
of the position b occupies is the sort of that which q occupies. 
85. The sorts of the letters and the sorts of the positions are both dealt with as 
units. Further, we regard each letter as belonging to a particular collection regarded 
in the aggregate as a single unit, say a unified collection. 
86. Thus a letter bears definite relations to 
(1.) its sort; 
(2.) the unified collection to which it belongs ; 
(3.) the sort of its position. 
Representation of Aspects of Collections by Arrays of Letters. 
87. Thus we have an exact copy in a different dress of the state of things considered 
in secs. 73-83. If then we represent 
(1.) the units of any collection by the sorts of letters; we may represent 
(2.) the sorts of places occupied by those units in a correspondence with an 
undistinguished collection by the sorts of the positions of the letters ; 
(3.) an aspect of the collection by a collection of letters in sorts of positions ; 
(4.) unified aspects by unified collections of letters ; 
(5.) aspects of single units by single letters. 
88. Taking, then, arrays as our collections of letters, an aspect may be represented 
thus abcdef . (i.e., in the manner adopted in sec. 7), where the order of the letters 
is to be regarded as material to this extent, viz., that though the aspect which is 
