Mr. T. S, Davies's Note on Numerical Transformation. S^l 



direction Jn .which the current travels depends on the relative 

 positions of the earth connexions, however circuitous may be 

 the roiite'of the" wire itself. 



I am, Gentlemen, 



Your obedient Servant, 

 Derby, April 12, 1849. W. H. Barlow, M.I.C.E. 



XLIX. Note on Numerical Transformation. 

 By T. S. Davies, Esq., F.E.S. L. Sf E.* ,, 



IN my notes on Mr. Cockle's paper (Phil. Mag., S. 3. 

 vol. xxxii. p. 351), it was incidentally suggested to express 

 the conjugate pair of roots of one of the quadratic factors of 

 an algebraic equation by « + /3 and a— 13, without assigning 

 the algebraical form of ^. It was, moreover, proposed to form 

 the two subordinate equations which contained relations be- 

 tween a and /3, as is now generally done, after the example of 

 Lagrange. 



It has been objected by analysts with whom I have corre- 

 sponded or conversed — analysts who would not have raised 

 frivolous objections to any proposition whatever — that by 

 the assumption of these forms I deprived myself of the means 

 of forming those two equations upon any legitimate principle ; 

 inasmuch as their derivation is founded entirely on the appli- 

 cation of the principle of incongruity, by showing that in 



X + Y/3^/^=0 



*ixie must have simultaneously 



X=0, andY=0. 



I may remark in the first place, that though I do not ques- 

 tion the legitimacy of this argument when all the roots are 

 imaginary, I still think it ambiguous when some of the roots 

 are real, and altogether fallacious when there are no imaginary 

 roots at all. 



And, in the second place, that any process founded on this 

 for the determination of the real roots of an equation, is totally 

 deficient of all legitimate foundation. 



I proceed, however, to the object of this note, which is to 

 form the equations X' = 0, Y' = on the unrestricted forms of 

 the pair of conjugate roots, « + /3 and «— 13. I use accented 

 letters, because the quantities represented by X' and Y' differ 

 in the signs of the alternate terms from X and Y. 



If/(,r) = be an algebraic equation of an even degree, and 

 a + jS, «— 13 two of its roots, we have 



* Communicated by James Cockle, Esq., M.A., Barrister-at-Law. 



