has as many Roots as if has Dimensions. 1 1 3 



ingan equatiou^^v is put for the unknown quantity, and is ope- 

 rated upon by the rules of the first part of algebra, the necessary 

 consequence is, that by the operations x acquires an algebraic 

 expression. Hence to include every form that it can have, we 

 may substitute for x in any equation /(a;) = 0, the general alge- 

 braic symbol z + yV — l. This substitution is legitimate also 

 in another respect, because when the equation is thereby trans- 

 formed into one of this form, 



<f>{z, f) + y .f{z, ,/) s/~l=0, 

 we must have separately, 



cf>iz,y^)=Oandy.f{z,f) = 0; 

 and thus there are two equations for determiuing the values of 

 the two unknown quantities z and y. The second of these equa- 

 tions vanishes if y = 0, in which case the other equation becomes 

 identical with the original equation. But the value y = applies 

 whenever the original equation has possible roots. This trans- 

 formation, therefore, does not help us to find the possible roots 

 of/(<r) = 0. As, however, this equation is by hypothesis nume- 

 rical, there are always means of finding the possible roots. For 

 instance, by substituting consecutive numerical values for x, 

 separated by sufficiently small differences, and extending far 

 enough positively and negatively, every possible value of x will 

 be detected, and may by the ordinary modes of a})proximation 

 be obtained as accurately as we please. When one such value « 

 is found, the equation may be reduced one dimension by divi- 

 ding by x — a. As the equation may contain equal roots, it must 

 be ascertained by trial whether the same quantity is a factor of 

 derived equations. The number of equal roots is thus found, 

 and the dimensions of the equation may be reduced accordingly. 

 The same process must be gone through for all the other possible 

 roots. Thus the residual equation, %i(j') = 0, will contain no 

 possible root, and be of lower dimensions than the original equa- 

 tion by the number of possible roots in the latter. U z + y v/ — 1 

 be now substituted for x in the equation ^,(.x')=0, and the two 

 resulting equations be 



<f>,{^,y')=0, y.f^{z,y^)=0, 



y can no longer be supposed zero, becrusc: in consequence of the 

 preceding operations, ;)5;,(.r)=0contauis no possible roots. Hence y 

 must have a possible value different from zero, and z a correspond- 

 ing possible value such that the two values satisfy the equations 

 <f)i{z, y"^) =0 and ^i{z, y^) = 0, at the same time ih'Ai z + y \/ — 1 

 satisfies the equation j^,(,2)=0. These are necessary conse- 

 quences of the legitimate assumption, that the x of every equa- 

 tion stands for an algebraic expression. Hence after eliminating 

 Phil. May. S. 4. Vol. 17, No. 112. Fob. 1859. I 



