1885.] A General Theory of Mathematical Form. 395 



refer to a number of separate units without reference to the various 

 " aspects " of the collection. Different aspects of the same collection 

 of n units will be spoken of as different w-ads. 



(9.) Every collection of units has a definite form, due — (1) to the 

 number of its component units, and (2) to the way in which the 

 distinguished and undistinguished units, pairs, triads, &c, are dis- 

 tributed through the collection. Two collections of the same number 

 of units, but having different distributions will be of different forms. 

 The angular points of a cube and of an octagon furnish examples of 

 two systems of eight units, having different distributions. In the 

 former case there are three sorts of pairs, in the latter four. 



(10.) Two collections which are undistinguished will be of the 

 same form, but two collections which are of the same form are not 

 necessarily undistinguished. 



(11.) Each of the forms which a system k of n units can assume 

 owing to varieties of distribution is one of a definite number of 

 possible forms, and the peculiarities and properties of the collection 

 depend, as far as the processes of reasoning are concerned, upon 

 the particular form it assumes, and are independent of the dress — 

 geometrical, logical, algebraical, &c. — in which it is presented ; so 

 that two systems which are of the same form have precisely the same 

 properties, although the garbs in which they are severally clothed 

 may by their dissimilarity lead us to place the systems under very 

 different categories, and even to regard them as belonging to " diffe- 

 rent branches of science." 



(12.) It may seem in some cases that other considerations are 

 involved besides " form," but it will be found on investigation that 

 the introduction of such considerations involves also the introduction 

 of fresh units, and then we have merely to consider the form of the 

 enlarged collection. 



(13.) In order to put form in evidence some " accidental " clothing 

 is of course necessary; if, however, we employ more than one species 

 of clothing, each species being uniform and suited to forms of every 

 kind, the disturbing effect will be reduced to a minimum. 



§§ 14—17. Units. 



The variety of the units which may be dealt with in an investiga- 

 tion is here indicated ; e.y., we may have a material object dealt with 

 as one unit, a quality of that object as another, a position it occupies 

 in space as a third, and a statement about it as a fourth. The diffi- 

 culty of specifying the units considered is referred to, and cases are 

 discussed in which mistakes are likely to occur. 



