Extensions of Quaternions. 129 



vector kinds : 



S(AVtV'-t"Va') = 0; (63) 



V(^Vi'^" + i'^VuO = ^"Sw'-tStV'; . . (64) 



because the law (32) of conjugation, iIt = Ku', gives, by (41), 



For the same reason, no essential change is made in either of 

 the two equations, (63), (64), by interchanging o and l" ; but if 

 we cyclically permute the three vector-units, 1 1' c", then (63) gives 



S(fcV6V') = S(t'Vt"0 = S(^"Vit'); • . . (65) 

 and there arise three equations of the form (64), which give, by 

 addition, 



Y(LYcU^' + o'Yi"L+L"yLi')=0; .... {66) 



and therefore conduct to three other equations, of the form* 



Y(tVL'o") = i"Scc'-c'SL"i (67) 



Equating t" to t, the two equations (65) reduce themselves to 

 the single equation, 



S{lVu') = 0', (68) 



and the formula (67) becomes 



V(iya') = tV-^Su': (69) 



both which results become identities, when we further equate o^ 

 to L. And no equations of condition, distinct from these, are 

 obtained by supposing l^ — i!, or l' = l, in (65) and (67). 

 The number of the symbols l being still supposed =^, and 

 therefore by [5.] the number of the constants which enter into 

 the expressions of their n^ binary products (including squares) 

 being =l(n^ + 7i), these constants are thus (if possible) to be 

 made to satisfy ^(n^—n) associative and scalar equations of con- 

 dition, obtained through (63), from the comparison of the scalar 

 parts of the two ternary products, i . iJJ' and i v' . a" ; namely, 

 n(n—l) scalar equations of the form (68), and ^n(n--l)(n—2) 

 such equations, of the forms (65). And the same constants of 

 multiplication must also (if the associative law is to be fulfilled) 

 be so chosen as to satisfy ^(n^—n^) vector equations, equivalent 

 each to n scalar equations, or in all to ^(n'^—n^) scalar condi- 

 tions, obtained through (64) from the comparison of the vector 

 parts of the same two ternary products (51) ; namely, n{n — l) 

 vector equations of the form (69), and ^n(n—l)(n—2) other 

 vector equations, included in the formula (64). This new ana- 

 lysis therefore confirms completely the conclusion of the fore- 

 going paragraph respecting the general existence of i(n'*— w^) 



* This formula is one continually required in calculating with quater- 

 nions (compare page li of the Contents, prefixed to the author's Lectures). 

 Phil. Mag. S. 4. Vol. 8. No. 50. Aug. 1854. K 



