Extensions" of Quaternions. 267 



and of which the comparison conducts to the four other distinct 

 conditions : 



r, Wj -n^r^= y^ + n^t^ = - /g^^ - t^u^'A '^ .^ 



r^u^ +mjrQ=y^—s^u^= — l^h—m^s^ ;J 

 where cyclical permutation of indices is still allowed. The equa- 

 tions obtained from the types (137) (138) would be found (as 

 the theory requires) to be merely consequences of these; for 

 example, by making e= 1, f— 2, y = 3, A=4_, those two types give 

 only the conditions, 



h{^i-^i)=Pi{r2-^^3)'^PA+Ps%y . . . (146) 

 which are obviously included in (143). 



[22.] With respect to those other homogeneous equations of 

 the second dimension, between the symbols [fffh], which are 

 obtained immediately, or without any elimination of the symbols 

 (/), {fff), from the general conditions of association, and are in- 

 cluded in the formula (96), they may now be developed as follows. 



Making k=f in (96), and then interchanging / and e, for the 

 sake of comparison with (123), we obtain the type. 



Ill- fgg '9ee-gjy.fee=T{gfh,hee); . . (147) 



which includes generally ^n[n — \)[n—%) distinct equations, of 

 71—1 terms each. For quines, we have thus 12 equations of 

 3 terms sufficiently represented by the following : 



n^nc^-m^m^=p^r,^, Wi^^ + mgW, = - /3W3 ; ~1 ^ 



r^mg + z^2 r 1 = ^2^3, r^n^ —u^r^ = -\- t^m^',J 



the value 4 being attributed to the index h or e, in forming the 

 equations on the first line, but to / or ^ for the second line. 

 For quadrinomes, the corresponding equations are only three, 

 namely 



Q=n^n^—m^mc^=n^nQ—m^m^=nQn^—mQmi; . (149) 



which however are sufficient, in conjunction wdth the three lately 

 marked as (140), to reproduce the six equations of the two last 

 lines of (86). In general, by adding and subtracting the two 

 types (123) (147), we obtain the formula, 



ifee±fgff){ffee+ffff) = T{fffh){hff+hee)-T{feh.ffhe) ; (150) 



where, as a verification, if we take the lower signs, and inter- 

 change / and g, so as to recover the first member with the upper 

 signs, the comparison of the two expressions for that member 

 conducts to an equation between the two second members, which 

 may also be obtained by the comparison of (125) and (127). 



[23.] Again, making /=:e in (96), and then changing k to/, 

 we obtain the formula, 



IV. 0=(eff/){eff+egff) + Xiegh .ehf); . (151) 



