396 Professor Sylvester on the Solution of 



q 2 -2bq + d = Q,qr + rq-2bq-2b iq + 2e=0, r 2 -2b 1 q + d 1 = 0, 



pq + qp-2bp—2fiq + 2e=*0, p*-2fip + 8=0, 



pr + rp — 2b x p — 2 fir + 2e l = 0] , 



and if we write 



a?-Bx + T> = 0, 



then _ :r + 8Dp-BD 



^"Sq-dBp + B'-D'' 



and I find by perfectly easy and straightforward work that B, 

 D may be determined by means of the following equations : — 



( B2 - p ) 2 + 2(b-fiB)^^- +(d-2eB + 4:BB 2 ) = 9\, 



WD — T3D 2 B 2 — D 



eJL^H + (& x + 3ffD) ^ " + (e-« 1 B + 3*D-68BD) = 3B\. 

 o o 



B 2 D 2 -2(6! + 3/3D)BD + d x + 6J)e 1 + 9SD 2 = J)\. 



The order (by which I mean the number of solutions of this 

 system of equations) is readily seen to be the same as that of 



■ D 3 +.B 3 D+-B 4 D =0 



.BD 2 +.B 3 D+-B 5 =0; 



i. e. is the same as the degree in B of B 3 (B 5 ) 2 . R, where R is 

 the resultant of 



- D 2 + ■ B 2 D+ . B 4 and - D 2 + ■ B 2 D-f . B 4 . 

 Hence the number of solutions is 3 + 10 + 8, i. e. is 21. 



Practically, therefore, we have now sufficient data to deter- 

 mine the number of solutions of a unilateral equation in 

 quaternions of any order co; for it is morally certain that such 

 number is a rational function of co ; and as it cannot but be 

 of a lower order than ft) 4 , we have only to determine a cubic 

 function of co whose values for ft) = 0, 1, 2. 3 are 0, 1, 6, 21, 

 which is easily found to be ft) 3 — co 2 + co; so that the evapo- 

 ration is a) 4 — ft) 3 + ft> 2 — co, i. e. (ft) 2 + l)(ft) 2 — w). 



Practically also we can solve (subject to hardly needful 

 verification) the number of roots of a unilateral equation of 

 the special form 



®» + q aP + q - l afi- l + ... + q o =O. 



For when # = &), we know the number is ft) 2 ; and when = 1, 

 the number is ft) 3 + ft) 2 — ft) ; consequently if the second differ- 

 ences of the function of (ft), 6) which expresses the number of 

 roots are constant, the value of this function when = co — 1 is 

 ft) 3 — ft) 2 + ft), w r hich we have found to be the actual number; and 

 consequently, if the second differences are not constant, they 



