Extensions of Quaternions. 283 



which, when we interchange g and h, reproduces the formula 

 (192); and shows thereby that the sub-type (127), included 

 under type II., is satisfied by our new type VI., which indeed it 

 had assisted to discover. The same equation (192) may also be 

 derived from the formula (205), by dividing each member of 

 that formula by (fee+/hh)(hff+hgg), and attending to the ex- 

 pressions given by type VI., for (egf) and (ge/i) respectively. 

 To interchange e, h, in (211), and divide, would only conduct to 

 another equation of the same form as (214). Permuting cycli- 

 cally the three indices e, /, g in (209), and multiplying together 

 the two equations so obtained therefrom, the product gives 



fhg .ghe = gff+gee _ 



ehg .fhe eff+egg' 



and if we multiply this equation by (209) itself, we find that 



ehf.fhg.ghe=ghf.ehg.fhe. . . . (216) 



In fact if we operate thus on the expression (197) for {ehf) , or 

 for its equal (ehf), or on the formula (196), we are led to this 

 new equation, 



ehf.fhg.ghe=(hee + hgg)(hff+hee)(hgg + hff), . (217) 



of which the second member does not alter, when we interchange 

 any two of the three indices e, /, g. Another multiplication of 

 three equations of the form (209), with the cycle egh, conducts 

 to the equation [/] =0 of (204). Interchanging e, h in (210), 

 and substituting the value so obtained for the product of the 

 two extreme factors of the second member of (217), we find this 

 other expression, 



ehf.fhg.ghe=hef.hfe.(hee + hff); . . (218) 



which is still seen to remain unaltered, by an interchange of e 

 and/. Interchanging/, g, and dividing, we obtain by (216) an 

 equation of the same form as (213)-; and if we divide each mem- 

 ber of (218) by {hef), we are conducted to the formula 



IV.. fhg.hge = hfe.{hee + hff), .... (219) 



which is of the same form as the equation (191), or as the type 

 IV., and may be changed thereto by cyclical permutation of the 

 four indices efgh. The same relation (219) may also be derived 

 more directly from type VI., by substitutions of the values (198); 

 fur it will be found that the definition (197) gives this identity, 



{MoW«)o=V«* + W)Wo- . . . (220) 

 The conditions of type IV., like those of type I., and of the sub- 

 type (127) of II., are therefore all included in those of the new 

 type VI, j which gives also in various ways this other formula 



