360 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION, ETC. 



is there proved, first, that after a rational function of n dimensions has attained a minimum value 

 corresponding to a real value of x, it is possible to diminish the function still further by assigning 

 to x an increment of the form h + ky/ - l, and then it is shewn that by assigning to x a value of 

 like form, it is possible to give to a rational function of x of even dimensions a series of 

 continually increasing or diminishing values, which propositions are akin to, but far less general 

 than, those which I have proved in Arts. (8) and (10). Nevertheless the mode of viewing the 

 subject is the same as that which I have adopted, and indeed suggested to me the possibility 

 of illustrating the theory of equations by reference to the curve of double curvature, which represents 

 the succession of real values of a function of x corresponding to values of the form .r + y v— 1 : 

 apart from which geometrical illustration, the theory of the roots of equations which depends upon 

 the demonstrated impossibility of a maximum or minimum value of f(x), when the values of x are of 

 the form x + y\/ - 1, appears to me to be more luminous than any other which I have seen. 



H. GOODWIN. 



