1864.] 



the Discovery of Imaginary Roots, 



181 



real, and for the other conjugate. This brings to view the necessity of 

 there being in general a theory for equations with conjugate coefficients, 

 which for greater brevity may be termed conjugate equations, analogous to 

 that for real equations in respect of the distinction between real and imagi- 

 nary roots in the latter. A conjugate equation is one in which the coeffi- 

 cients, reckoning from the two ends of the equation, go in pairs of the form 

 P i iq_i with the obvious condition that when there is a middle coefficient 

 this must be real. Such an equation may be supposed to be so prepared 

 that, when thrown into the form P + 2Q, P and Q shall have no common 

 algebraical factor ; and when this is effected, it may easily be shown that 

 the conjugate equation can neither have real roots nor roots paired together 

 of the form X+z'/z, X—i^-i respectively. How, then, it may be asked, is the 

 analogy previously referred to possible ? On investigation it will be found 

 that the roots divide themselves into two categories, each of exactly the 

 same order of generality, — viz. solitary roots of the form e*^, and associ- 

 ated roots which go in pairs, the two roots of each pair being of the form 



pe^y — respectively ; so that, following the ordinary mode of geometrical 

 9 



representation of imaginary quantities, the roots of a conjugate equation 

 may be denoted by points lying on the circumference of a circle to radius 

 unity (corresponding to solitary roots), and points (corresponding to the 

 associated roots) lying in couples on different radii of the circle at reci- 

 procal distances from the centre, each couple in fact constituting, accord- 

 ing to Prof. W. Thomson's definition, electrical images of each other in 

 respect to the circle. If the circle be taken with radius infinity instead of 

 unity (so as to become a straight line), then we have the geometrical 

 eidolon of the roots of an ordinary equation, the solitary roots lying on a 

 straight line, and the associated or paired (imaginary) roots on each side of, 

 and at equal distances from the line. 



In the inquiry before us, whether the variable point belong to the real 

 or conjugate part of the plane of the D curve, it is shown to remain true that 

 the number of associated roots will be two, if it lie inside the curve, and four 

 if it lie outside. The author then suggests a probable extension of Newton's 

 rule to conjugate equations of any degree. In conclusion, he deals with a 

 question in close connexion with, and arising out of the investigation of 

 this rule, relating to equations of the form ^ + {ax + b)^ = Q, to which, 

 for convenience, he gives the provisional name of " superlinear equations" 

 (denoting the function equated to zero as a superlinear form), and esta- 

 blishes a rule for limiting the number of real roots which they can con- 

 tain, which is, that if such equation be thrown under the form 

 \{w + cy' + \{x-\-c^Y^- .... +X„(^ + c„)"^=0, 

 and . . . c„ be an ascending or descending order of magnitudes, the 



equation cannot have more real roots than there were variations of sign in 

 the sequence Xj, . • • Am> (— )"'Xi. 



