a Class of Equations in Quaternions. 397 



must be sometimes positive and sometimes negative, which is 

 in the highest degree improbable. Hence in all probability- 

 it will be found that the required number of solutions in the 

 form supposed is (1 + 0)oo 2 — 0co. 



I need hardly add that the nine quantities 2b, 2& 1? 2/3 ; 

 2e, 2e x , 2e ; d, 8, d 1 , which occur in the discussion above given 

 of the general unilateral cubic, or, say, rather the ten quan- 

 tities obtained by adding on to these unity, are the ten coeffi- 

 cients of the determinant to the binary matrix (x+py + qz + rt) 9 

 which of course there is not the slightest difficulty in ex- 

 pressing in terms of scalar and vector affections of p, q, r 

 and their combinations, if any one chooses to regard them as 

 given in quaternion form. 



Scholium. — In what precedes it is very requisite to notice 

 that only general cases are considered ; and that there are 

 multitudinous others which escape the direct application of 

 this method, and do not conform to the rule which assigns the 

 number of solutions. Thus, ex. g?\, the equation x 2 +px = 0, 

 besides the solutions x = 0, x = — p, will have two others which 

 will require the method of the text to be modified in order to de- 

 termine. Or take the most elementary case of all, the simple 

 equation px = q. If p is not vacuous (i. e. if its determinant when 

 regarded as a matrix, or its modulus when regarded as a quater- 

 nion, is finite), there is the one solution x=p~ l q. But if p is 

 vacuous, then, unless q is also vacuous, the equation is insoluble. 

 If £ = 0, there will be two solutions; one of them #=0, the other 



x = conjugate of pin quaternion terminology; or x= o*—d? 

 when p = I t in the language of matrices. If, p still remain- 

 ing vacuous, q is vacuous but not zero, a further condition 

 must be satisfied; viz. if p= j j, and q = \ g, the condition is 



aS + ctd — by — c/3 = 0; 

 or if p = a + bi + ij + dk and q = * + (3i + yj+ 8k, the condition is 



ax + bp + cy + d8 = 0. 



When this condition (besides that of q being vacuous) is 

 satisfied, the equation px = q is soluble, and p~ l q becomes 

 finite but indeterminate, containing two arbitrary constants*. 

 So in general if p, q be two simply vacuous matrices of any order, 

 the condition that the equation px=q may be soluble, or, in other words, 

 that p-iq (a combination of an ideal with a vacuous matrix) may be non- 

 ideal, may be shown to be that the determinant to the matrix \p+pq 

 (where X, /x are scalar quantities) shall vanish identically — which (p being 

 supposed already to be vacuous) involves just as many additional con- 

 ditions as there are units in the order of the matrix. 



