on Quadruple Algebra, and on the Theory of Equations. 437 



we see that, when two unknowns are connected by a quadratic 

 equation, then any higher function of those unknowns admits of 

 material simplifications depending upon the transformation of 

 the quadratic to the form (/). Thus, a cubic function may be 

 made to take the form 



x 3 + a 3 !/ 3 + S 2 # 2 + c 2 y 2 + dx + ey +/, ...(«) 

 or that indicated by the relations 



dx= — da~ 1 x z y, ey = — ea~ 1 xy*, 

 or the following form, — 



(x + ai/) 3 +(bx±cy)* + d'x + e'y+f, . . . (v) 



besides others. It may perhaps be worth while to inquire 

 whether, by sacrificing the symmetry of (/), the cubic function 

 may not be rendered symmetric ; and also to examine the cubic 

 system, corresponding to (h) and (i) of p. 292, vol. ii. S. 4, which 

 may be obtained when the latter function is supposed to vanish ; 

 and to ascertain whether, as the solution of two simultaneous 

 quadratics may be made to depend on that of a cubic, so the 

 solution of a simultaneous quadratic and cubic may not be made 

 to depend on that of an equation lower than the sixth degree. 

 The same observations apply when in place of a cubic we have a 

 higher function. I think it right to add, that I arrived at the 

 solution of (n) and (o) (Ibid. p. 293) by an assumption of the form 



x='kx' + /j,y' + r ) 



and then seeking, by this linear transformation, to render the 

 resulting system pseudo -homogeneous — that is to say, free from 

 the first dimension of x' and y'. The expression (p) for y was 

 at once indicated, and, thence, my solution of the given system 

 — under which the two members become identical : x and y each 

 satisfying a functional equation of the form 



f(x)-x=g. 



Compare this with the functional relation satisfied by the roots 

 of a certain trinomial quintic (S. 3. vol. xxxii. p. 52). 



(b.) There is an abnormal system of quadruple algebra in 

 which the imaginaries are subject to the conditions 



-« 2 =^ 2 = 7 2 =1, 



«/3= — /3«=y, 7«= — «7=/3, 7/3= — yS7 = «. 



This I have proposed to call the coquaternion system (see p. 434 

 of vol. xxxv. S. 3.). Let M be the modulus of the coquaternion 

 A, then we may make 



A = M (« + ut-\-/3» ->- --V 



