On the Solution of a Class of Equations in Quaternions. 393 

 From the last two equations, eliminating v, there results 



(2Xu-2bX-/3) 2 -4:X(2Xu-2bX-l3)u + (y-4.cX)u 2 = 0. 

 Hence substituting X — a for u 2 , 



(4\ 2 + 4cX-7)(X-a)-(26X + /3) 2 = 0. 

 We have thus six values of u, viz. 



±\/X — u 



(where X has three values), to which correspond six values of 

 v, viz. 



V A, — ot 

 and, finally, 



2<£ = (p— -u + b)~ l {q — v + c) 



= ((p + bf—u 2 )- 1 (p+b--u)(q + c--v), 



or 



__pq + (c—v)p + (b — u)q + (b—u)(c—v) 

 X ~ 2(6 3 -rf-w 2 ) ~ ; 



which equation gives six values for x, and shows that ten have 

 evaporated. 



It is easy to account a priori for the solution depending 

 only upon a cubic in u 2 . 



For x 2 — 2px + q = is the same as y 2 — 2py + q = 0, where 

 y——x — 2p. But obviously, from the nature of the process 

 for determining them, B and are independent of the side of 

 the unknown on which the first coefficient lies. Hence the 

 actual B will be associated with B', W being what B becomes 

 when x becomes — x — 2p, which is obviously — B — 2b. 



Hence with any value of B + b which is u is associated a 

 corresponding — B— b which is — u. 



I will now proceed to apply a similar or the same method 

 to the trinomial cubic equation in quaternions (or binary 

 quantity) x^+px— q = 0, with a view to ascertain the number 

 of its roots. 



Retaining the same notation as before, and still supposing 



^ 2 -2B^ + D = 0, 



we obtain 

 and 



^ 3 + (D-4B'>+2BD = 0, 



:g + 2BD » 

 ~p + 4B 2 -D • 



* I use : and =4 to signify M *L and LM } respectively. 



MM. 



Phil Mag. S. 5. Vol. 17. No. 107. May 1884. 2 D 



