28 
MR. A. B. KEMPE OR THE THEORY OF MATHEMATICAL FORM. 
164. If we then take another correspondence not included among the former, we 
shall get a second collection of correspondences equal in number to the former, and so 
on until the correspondences are all exhausted. 
165. For example the six graphical units of the system of fig. 26 constitute two 
independent systems of the same form. Starting with the correspondence ^ which 
we may represent as a unit P, we may add correspondences °^ and ^ which we may 
represent by units Q and R respectively. Here we have only correspondences such 
as those referred to in sec. 163. It is to be observed that we cannot properly say 
that P represents any particular one of the three correspondences, for they are undis¬ 
tinguished ; the three units P, Q, R, together represent the three correspondences ; 
but if P is regarded as representing any definite correspondence Q and R will each 
represent definite correspondences, for P, Q, R, are each unique with respect to each 
other. If we now consider the correspondence . in which we have components of 
dje 
the same form corresponding as before, we get the correspondences ^ and . We 
may represent these by the units L, M, N. The whole system of units considered is 
represented by the following table — 
a b c d ef P Q R L M N 
a b c e f cl Q R P N L M 
a b c f cl e R P Q M N L 
b c ad ef RPQNL M 
b c a ef d P QRMNL 
fcca/dcQRPLMN 
c a b cl e f Q RPMN L 
ca&e/dRPQLMN 
c a b f d e P QRNL M 
166. We may of course consider self-correspondences of a system S in which com¬ 
ponents of like form correspond which are distinguished from each other. Thus in 
the case of the system a, b, c, cl, e, of fig. 2 7 we' may consider correspondences such 
