on Quadruple Algebra, and on the Theory of Equations. 439 

 7 ) are given by the formula 



Consistency must be observed in the sign of the radical p : hence, 

 considering <y % as positive, and giving to J^ 2) its original quadratic 

 form, if we make 



/s(9) = JW + R, (21) 



and represent by R^ q the coefficient of z z in R, and by y' T and 

 y" r the two values of <y r corresponding respectively to the positive 

 and negative values of p, the following relations obtain, — 



T P,q S = r y'pyq + VqVp + %,li ■ • • • (22) 



where, when p and q are equal, the middle term is suppressed. 



The subject of impossible equations has been pursued with 

 great assiduity and zeal by Mr. Robert Harley, who has discussed 

 it in the memoirs of the Manchester Literary and Philosophical 

 Society (vol. ix. pp. 207-235), and also in the Mechanics' Maga- 

 zine (vol. I.). He has shown that if 



Vx + \/^+l=0, then^=(-l) 2 {l-(-l) 2 } -1 • 

 This equation I had previously (Phil. Mag. S. 3. vol.xxxvii.p.283) 

 stated to be insolvable. So far, therefore, as that equation is 

 concerned, the views which I took in the paper last cited must 

 be modified ; but, as at present advised, I do not conceive that 

 they require any further modification. The existence of impos- 

 sible equations cannot but be admitted, until some satisfactory 

 and consistent mode of affecting 



\/m{ + l)* + n(-l) 2 or V p 



with an appropriate sign shall enable us to satisfy such equa- 

 tions as Y = and Z = 0. Consider for a moment the equation 

 Y = 0; its root x is subject to the conditions 



#-4=( + l) 2 , and*-l=4(-l) 2 , 



which can only be simultaneously satisfied by supposing that the 

 4 in Y = is of the form 4(-l) 2 , and the 1 of the form ( + 1) 2 . 

 What justifies these suppositions ? What prevents their violation 

 by the express form which we are at liberty to give to our sym- 

 bols ? It would seem that we should thus obtain an absolutely 

 impossible equation. Is there any law of affection by which, x 

 being taken of the form p, such difficulties can be obviated ? 



2 Pump Court, Temple, 

 April V.), 1h:,l\ 



