MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
53 
302. An algebra may be self-distributive, i.e., such that 
( a(bc )) = (( ab)(ac)). 
(See sec. 349.) 
303. An important class of primitive algebras is that in which ( ab) f a or b, 
except in the case of one unit z, which is such that (zz)=z, and also such that what¬ 
ever unit of S a may be [za)—a and (az)=a. Here every unit a is accompanied by 
another a such that (aa')=z. Ordinary addition and multiplication furnish two such 
algebras. In the former (ab) is a-\-b, a is 0, and a is — a ; in the latter (ab) is axb, 
z is 1, and a is 
304. The algebras which are such that in the case of each unit a we have (eta) = a, 
are also of considerable importance. We have such an algebra in the case of 
ordinary logic (sec. 360). 
305. We may in ordinary parlance speak of three units a K , & M , c v , composing the 
triad giving rise to a primitive equation, as if they wei’e units a, b, c, of S, without 
any ambiguity arising. The unit a when dealt with as a multiplier will be a K , and so 
on ; but it will not be necessary to be continually pointing this out. We may thus 
speak of products (aa) when there is really only one a. 
306. If in any system of units S every pair is accompanied by one or more units 
unique with respect to the pair, we may select one of these accompanying units in the 
case of each pah' and term it the product of the pair. Where there is only one such 
accompanying unit in the case of each pair, this will be the product, but if there be 
several such, any one of these might be chosen as the product. The most natural 
mode of selection in the case of undistinguished pairs is to select products which will 
with the pairs make undistinguished triads. (Secs. 343, 349.) 
307. In the case of distinguished pairs we cannot of course do this, but we may in 
some cases choose products so that the resulting triads will all be of the same form, 
this not being so if other products be chosen ; or we may choose the products, so that 
by ignoring some difference the triads all become undistinguished from each other, the 
products still being unique with respect to the pairs. 
308. In cases in which some pairs are not accompanied by a unit unique with 
respect to them, or even in other cases, we may add to S a single system of one unit 
Z, and call this the product of the pairs; Z being also considered as the product 
of each unit of S when multiplied by or into Z. (Sec. 3 19, and cf. sec. 343.) 
309. In certain cases the products considered may not be unique with respect to 
the multiplier and multiplicand, but with respect to a collection containing them 
and certain other units, which remain the same in the case of every multiplier and 
multiplicand of the system, and may therefore be termed constants. 
An instructive example of this is furnished by a system of collinear points. If i, z, 
u, a, b, be five points of such a system, there is one point, and one only, which is such 
that the six points 
