Extensions of Quaternions. 285 



we are in this way led to establish the following seventh type * for 

 quines : 



VIL. fgh.hff=v f .gee+fhh.gff . . . (228) 



Or since, by the sixth type, we have already 



VI. . fgh . (hee + hff) = (fee +fhh) (gee +gff), (229) 



it is only necessary, for the purpose of satisfying the conditions 

 of type II., or the equations (143) (145), to suppose besides 

 that 



III.. fgh.hee=fee.gff-fgg.gee; . . (230) 



such being the expression which remains, when we subtract 

 (228) from (229). But this last equation (230) is precisely 

 what the type III., or the formula (147), becomes for quines ; 

 it reproduces therefore the equations (148), with a correction 

 elsewhere noticed (namely the substitution of s 2 ?j 3 for s 2 r 3 ) : and 

 conversely, if we retain that old type III., it will not be necessary, 

 although it may be convenient, to introduce the new type VIL, 

 in combination with type VI. And if in (230) we substitute 

 for the symbol (fgh) its value given by (229), and so combine 

 types III. and VI., we obtain the equation 



fee +fJih _ gff. (fee +fgg) -fgg . (gee +gff) 

 hee + hff kee.(gee+gff) ' ' ^ 



that is, by (209), 



f91 + f9l_g_ff..f]HL. f 232) 



hge hee hee ghe' * ' 



or finally, 



V.. hee.fge=fgg.ghe-gff.fhe. . . . (233) 



But this is exactly what the type V., or the formula (154), 

 becomes for quines, when we suppress the sum £", change k 

 to h, and advance cyclically the three indices efh ; it includes 

 therefore the equations (155), which were the only remaining 

 conditions of association to be fulfilled. If then we satisfy the 

 two types, III. and VI., we shall satisfy all the conditions of asso- 

 ciation for quines : since we shall thereby have satisfied also the 

 four other earlier types, namely those numbered as I. II. IV. V. 

 It only remains then to consider what new restrictions on the 

 constants (eff) are introduced by the comparison of the values 

 which type III. gives for the other constants (efg), as expressed 

 in terms of them, with the values furnished by type VI. ; or to 

 discuss the consequences of the following general formula, ob- 



* It will be shown that this single type (the seventh) includes all the 

 others, or is sufficient to express all the general conditions of association, 

 tut ween the L' 1 symbols of the (onus (<•;/}/) and {eff). But the eliminations 

 required for this deduction cannot be conveniently described at this stage. 



