450 The Rev. T. P. Kirkman on Pluquaternions, 



that can be formed with seven things, so as to have no duad 

 in common, is seven. Of these, there will be three in which 

 a Q is found ; let the three be made all of one sign, by the in- 

 version of a duad if necessary, a change which can alter no 

 value, since — a b Q c = + a Q c Q b . Neglecting subindices, let 

 the three triplets be abc, ade, qfg. We shall consider the order 

 abcdefg to be alphabetical, and shall inquire what the remain- 

 ing four triplets will be, when written out in the same order. 

 Because 



abc=ade— + 1 ; 



be = de, 

 and —c=b.de. 



Now b.de, whether real or not, if b, d, e be any three imagi- 

 naries, cannot differ in sign from e.db; for if it does, 



b.de=—e.bd=d.eb——b.de, Q.E.A. 

 Therefore 



— c=b.de = e.bd, 



—ec =.—bd, 



or 



bd—ec, 

 and for the same reasons, 



be—cd. 

 Hence, in each of the following lines, any two equations are 

 implied in the third : — 



bc=de, be—cd, bd=—ce;' 



oc=fg, hg^cf, bf=-cg;l . . . (A.) 



de=fg, dg=ef, df=-eg.^ 



It is necessary that three similar lines deduced from any 

 three triplets of the system, beginning with another imaginary 

 than a, should be consistent with these. Now it is readily 

 seen, that the system of seven triplets, of which three are abc, 

 ade, qfg, must perforce contain either bdf or ceg; and the 

 reader will convince himself with less pains than the perusal 

 of a demonstration would cost him, that the above conditions 

 can, and thus only can, be fulfilled, by giving to that triplet 

 of these two, which is found in the system of seven, a sign 

 contrary to that of the remaining three. Hence all possible 

 arrangements, if abc, ade, qfg are in the system, are comprised 

 in the two following types, each of four systems : — 



