Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION, ETC. 34.'J 



.3. If we wish to represent the changes of value of f{x) taking into account only real values 

 of .r, the mode adopted would be to construct the curve defined by the equation 



^=/(*) (1). 



but if we wish to give a general representation of the changes of the function, taking into 

 account both real and imaginary values of a?, we must construct the locus of the equation 



z^f(.v + yx/^) (a), 



where x y and z are to be considered as co-ordinates of a point in space as is usual. Now if 

 we restrict ourselves to values of z which are real, equation (2) will divide itself into two 

 equations, which will be the equations of a curve of double curvature, and the points in which 

 this curve meets the plane of wy will determine by their distances from the origin the roots 

 of the equation f{v) = 0. 



I will observe here that f {v) will be considered throughout this paper (unless the contrary 

 is stated) as the representation of the quantity 



x' + p^ ,1."-' + p^x"-- + + p„, 



where P\Pi P„ are real and either positive or negative. 



4. The two equations to which (2) corresponds may be expressed in several ways, which 

 I shall here put down together. 



By direct expansion, equating real and imaginary parts, and dividing the second equation 

 by y, we have 



z=f{.v) -/"(,.) j^ +/- Cr) -^ - &c. 



=/ (r) -/" (^o j^+r i^) 1^ - &c. 



If n be even and = 2 m, these equations become 



^=/(*)-,r(*)^+/"w-^- + {-irr"' 



and if w be odd and = 2ni + 1, they become 



.(3). 



(4) 



^-f{^^)-f"{^v)j^+f"i-v)jl - + (-l)"'(2m+ \..v + p^)y-'» J 



o=f'{^v) -f"'(.v) 1^' +r{x) y~ - + (- lyf j 



The equations also admit of a very neat symbolical expression, thus*: 



, (5). 



" The method which I have given of representing the locus 

 of the efjuation z~f{.t) taking into account values of .r not 

 lying in the real plane, in applicahle mutatis ynutandis to curves 

 defined by an implicit relation between the co-ordinates. Thus, 

 let the e<|uation he 



f{x.z) = a^ {A), 



then pulling for x x + yV-l, thin bccomea 



which is equivalent to the two following, 



