and Homoid Products of Sums ofn Squares. 4-51 



of which eight systems any one will satisfy the test invented 

 by Mr, Cayley, of which mention has been made, and which 

 is proved by the preceding considerations to be both neces- 

 sary and complete for a system of seven triplets. 



The congruity of any of these, e. g. of those in the first type, 

 may be tested by taking some other order for alphabetical ; 

 for instance, 



which is still of the same type as before. 



One of these eight systems, being adopted, will furnish the 

 non-linear conditions among the seven imaginaries, by the aid 

 of which the product of two biquaternions is reduced to a bi- 

 quaternion. The system of the first type, having the two 

 upper signs, for example, expresses that 



and 



rf o = *o a o~ -fo K=9o c o> 



Hence in the biquaternion Q fl// , formed by this system, 



%maw 4 + ina, + {bc,—eblj + (de^ed,) + (fgi-gfih 

 and 



d u = dm, + ivd, + {ea, - ae,) - [fb, - bf ( ) + (gc, - eg,) ; 



for these are exactly the terms affected by a Q and d Q in the 

 product Q a Q ar But if we adopt the system of the second 

 type which has the two lower signs, we should form the same 

 functions thus, 



a w 5Baw, + wa^ibc^cb,) - {de,-ed^ - {fg-gj,) 

 d ll =dw l +>wd l -{ea i -ae^-(gb-bg^-(fc-cfy, 



where the given numbers wrw ; aa t . . .gg, are in both cases the 

 same. 



It thus appears, that a biquaternion QL may be formed, that 

 shall be equal to the product of any two Q a and Q a/ , in at least 

 eight different ways ; and the eight different values of the pro- 

 duct shall be equally congruous and consistent with our defi- 

 nitions ; the real quantities in Q„ and Q tt/ being any numbers, 

 and remaining unchanged in value, sign, or order. 



Let Q a and Q fl/ now be pluquaternions of more than seven 

 imaginaries. If Q a Q a/ =Qo //} there must be some complete 

 system of triplets into which the {6m + 1) or (6m +3) imagi- 

 naries are thrown. Let a c d e and aof Q g Qi both of one sign, 



