Cambridge Philosojyhtcal Society. 367 



Equations." By the Rev. H. Goodwin, late Fellow of Caius Col- 

 lege, and Fellow of the Cambridge Philosophical Society. 



The changes of value of any function of x, f {x), may be very 

 clearly, and for some purposes very usefully represented, by tracing 

 the curve defined by the equation z=f {x) ; and the positive and 

 negative roots of the equation / (.r)=0_will be the distances from 

 the origin at which the curve cuts the axis of x. 



In this memoir a similar method is apphed to the representation 

 of the changes of value of a function of {x), corresponding not only 

 to real values of x, bu^lso to values of the form x+y \/ —\. If we 

 make z=f {x+y V — 0' ^"^ restrict ourselves to real values of z, 

 the equation separates itself into two, which, it is shown, may be 

 represented symbolically by 



and = sin / y 4", ) / ('^)' 



and these will correspond to a curve of double curvature, the inter- 

 sections of which with the plane of xy will determine by the distance 

 of those points from the origin the imaginary roots of the equation 



The properties of this curVg ^^re fully ^jscussed for the case of/ {x) 

 being equivalent to .t"+;j, Xn.l\-{-Pi Xn-2+ • • • • +Pn, where ;>, p^ 

 . . . . pn are real ; and the following results are obtained. 



1 . The ordinate of the curve admits of no maximum or minimum 

 value. . 



2. The curve goes off into infinite branches, which lie in asymp- 

 totic planes equally inchned to each other, and which tend alter- 

 nately to positive and negative infinity. 



3. Any plane parallel to the plane of xy cuts the curve in n points 

 and no more. 



From this last result the existence of n roots and no more tor an 

 equation of n dimensions is the immediate result. 



Several well-known theorems are deduced from this view of the 

 subject, and are given as illustrations. 



The actual curves are traced, corresponding to the various cases 

 of the quadratic, the cubic, and the biquadratic equations, and to 

 the equation .r" — 1=0. 



In the conclusion of the memoir it is remarked that the results 

 obtained are not exclusively applicable to the case of algebraic 

 equations, and the methods are applied to the case oif{x)-sm x. 



The author trusts that the contents of this memoir, though not 

 adding to the number of known theorems, may yet be usctul as 

 putting the subject in a new light, and as furiushuig a method ot 

 demonstrating the existence of the roots of algebraic equations more 

 simple and direct than any other which he has seen. 



