MR. A. B. KEMPE ON THE THEORY OP MATHEMATICAL FORM. 
21 
alternative correspondences or self-correspondences, in others it may partially restrict 
them, in others it may exercise no controlling effect whatever, so that collections may 
go through all correspondences and self-correspondences with as much freedom as if 
there were no correspondence already existing. 
118. If two collections are such that each can go through all its self-corre¬ 
spondences while the other remains identically-correspondent, they will be said to be 
independent. If two collections are not independent, they will be said to be related. 
119. If to and to' be the number of self-correspondences of two systems S and S' 
respectively, the table representing the joint system S, S' will have mm rows if S 
and S' are independent. 
120. In the self-correspondences of a system every correspondence of undis¬ 
tinguished components occurs; but it is not of course in general the case that in 
the self-correspondences of any collection all the correspondences of undistinguished 
components of that collection occur. For example, in sec. 100 the only self-corre¬ 
spondence of a, b, c, is given by the constituent ^ ^, and here no correspondence 
of the undistinguished units a, b occurs. 
121. When investigating the correspondences of a number of (n + m)-ads we 
may for some time be occupied with the consideration of correspondences in which 
m of the units always remain identically-correspondent. The absence of change 
in the correspondence of the m-units may lead us to forget or overlook the fact that 
we are considering correspondences of the (to + n)-ads, and we may suppose that we 
are dealing with roads only. When then we find that a certain correspondence of 
two n -ads apparently does not exist, we must look closely to see whether we are 
not really considering correspondences of (n -f- m)-ads, and whether a change in the 
correspondence of the m units may not lead to the correspondence of the n which 
is supposed not to exist. 
122. Units which in any correspondence of two undistinguished aspects are 
identically-correspondent may be termed the foci of the correspondence. Three or 
more undistinguished aspects such that the foci of the correspondences are the same 
in the case of each pan, may be said to be confocal. We may also use the term 
confocal as applicable to the case of corresponding components of the corresponding 
confocal aspects. 
123. In cases in which, as mentioned in sec. 121, we consider (n + to)- ads as if they 
were -roads, we really pass from the consideration of the original units to that of 
other units which are arrived at by taking the to foci with each of the other units 
successively, and then regarding the resulting (to -f- l)-ads as single units. 
124. Differences between things can always be ignored, and thus we may at one 
time regard two collections of units as distinguished, and at another, by ignoring 
differences between them, as undistinguished. Here we are dealing with certain 
units, upon which the differences depend, in addition to those of the two collections. 
