MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
47 
The component simple networks of this group, viz. :— 
(ux), (ux'), (uy), (uy '), [uz), ( uz ), (uu), (mi), 
(uX), (uX') } (uY), (uT), (uZ), (uZ'), («U), (ttU'), 
are subject to the same laws of multiplication as the bi-quaternion expressions 
i, i, j, j, Jc, 1c, 1, 1, 
Coi, —C oi, 0)J, — COJ, (Jjh, —C ok, 0), — OJ. 
R-adic Groups. 
270. We may have a set such that the unified component r-ads constitute a 
group. We may call the set an r-adic group. If r— 1 the set is an ordinary 
group. 
271. In an r-adic group any r-aci may be made to correspond to any other r-ad, 
but the correspondence of two r-ads completely determines a self-correspondence of 
the set. Thus if the form of the set be known, any self-correspondence is fully repre¬ 
sented by a two-row r-column constituent of the table. 
272. In an 7*-adic group of n units the tabular representation has rows. For 
I n '-. 
all (r—l)-ads are undistinguished, and there are p of these, and while any one 
remains identically-correspondent the remaining (n— r+ l)-ad has self-correspondences 
characteristic of a group, i.e., has (n—r —1) self-correspondences. Thus there are in all 
(n—r— 1) — ,- A ■ self-correspondences of the group including the identical 
correspondence. 
273. A correspondence of two undistinguished aspects of an r-adic group has not 
more than r—1 'foci (sec. 122). 
274. Let S be a system of which the units are the whole collection of points lying 
on a straight fine, viz., a, b, c, d, . . . . Any aspect abed ... of S is what is usually 
termed the “range abed . . . .” For the range abed . . . we can by a homographic 
transformation substitute another range of the same points a, b, c, d, . . . , i.e., 
another aspect of S. Employing all the various homographic transformations we get 
a set of aspects of S. Now, if we assume that this set is a single system, i.e., that 
all aspects of S derived from abed ... by homographic transformations are undis¬ 
tinguished from abed . . . and each other, but are distinguished from all aspects which 
cannot be derived from abed ... by such transformations; then S will be a triadic 
group. For we can by a proper homographic transformation substitute for any three 
units of S any other three units of S, but when this substitution is made, every other 
unit of S has a definite unit of S substituted for it. 
