1885.] A General Theory of Mathematical Form. 397 



when two undistinguished aspects are regarded as corresponding unit 

 to unit. The possibility of representing any conceivable system by a 

 diagram consisting of graphical units and plain lines, or links, only is 

 shown. 



§§ 84 — 86. Letters, their Sorts and Positions. 



In arrays or other assemblages of letters each letter is of a parti- 

 cular sort, and occupies a position which is of a particular sort. The 

 nature of the collections of units here dealt with is discussed, and 

 they are shown to be the same as those considered in the case of 

 aspects of systems in general. 



§§ 87 — 88. Representation of Aspects of Collections by Arrays of 



Letters. 



Hence the propriety of doing that which is indicated by the heading 

 of these sections. 



§§ 89 — 99. Elementary Properties of Aspects. 



A variety of propositions about aspects are here given, which are 

 needed in subsequent parts of the memoir. Perhaps the most 

 important is the following : — 



" If abed .... is undistinguished from pars . . . . , and if I, m, n, o 

 .... be units other than a, b, c, d, . . . , there must be units w, x, y, z, 

 .... other than p, q, r, s, . . . . such that abed .... Imno .... is 

 undistinguished from pqrs .... wxyz . . . . " 



It is pointed out that if any aspect of a whole system S be given, 

 and all other aspects of S which are undistinguished from the given 

 aspect, the form of S is completely determined. 



§§ 100 — 111. Tabular Representation of Systems. 



Hence we get a simple and uniform method of representing any 

 system, viz., that of arranging the arrays of letters representing the 

 undistinguished aspects one above another in rows, so that letters 

 occupying " the same position " in the different rows all lie in the 

 same column. In the resulting table the order of the rows or columns 

 is immaterial. Various points about these tables are considered. A 

 table representing a single heap of n units will have | n rows ; and 

 one representing a discrete heap of n units will have one row only. 

 In the case of any other form of a system of n units the number of 

 rows will be intermediate between 1 and | n. 



§§ 112 — 129. Correspondences of Undistinguished Collections. 



These sections go somewhat fully into the correspondences of undis- 

 tinguished collections indicated by the tabular representation of a 

 system. The nature of the correspondences where two component 



