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XVI. On the Connexion of the Circle and Hyperbola ; and 

 on the Geometrical Interpretation of Imaginary Exponen- 

 tials. By H. S. Wauner, Esq.* 



HPHE following investigations relate to an interesting branch 

 •*■ of modern science ; but as it is dangerous to claim, at 

 this time, anything as one's own without a careful examina- 

 tion of preceding writings, and as I am not acquainted with 

 the recent progress of research in regard to what were called 

 imaginary quantities, I send these observations for publica- 

 tion, merely on the chance that they are new; hoping that if 

 they are not so, I may be informed of the fact through the 

 columns of the Philosophical Magazine. 



1. The assumption on which I proceed is that, — 



The proper geometrical interpretation of' A*/ — 1 is a line 

 of the length A, measured in a direction perpendicular to that 

 in which it 'would have been measured if it had not been multi- 

 plied by a/ — 1. 



2. Taking the equation of the equilateral hyperbola (the 

 centre being the origin, and, for simplicity, the semi- axis 

 being the unit of length) y^=x 2 — l ) we find, that for a value 

 of x (positive or negative) greater than unity, y is possible, 

 determining the opposite hyperbolas ; but that for a value of 

 x less than unity, the equation assumes the form ?/ 2 = — 1 

 x (1 — x 2 ) or y = V — 1 x V{\ — x 2 ). Hence (y= Vl — x 2 

 being the equation of a circle whose radius = 1) by the as- 

 sumption above, the hyperbola will be represented between 

 x= + l and ,r= — I, by a circle (radius 1) whose ordinates 

 are measured perpendicular to the plane of xy; that is by a 

 circle standing upright, or a circle described on a plane at 

 right angles to the plane of xy, the intersection of the two 

 planes coinciding with the axis of x. The perpendicular 

 plane I must, for want of a better name, call the imaginary 

 plane. 



3. Considering the circle in the same way, we shall find 

 that for positive or negative values of x greater than unity, it 

 is represented by a pair of opposite hyperbolas, drawn on the 

 imaginary plane. Hence we arrive at this remarkable result, 

 that the equilateral hyperbola and the circle are similarly re- 

 lated to planes at right angles to one another, or that to change 

 the one curve into the other, we have merely to take the imagi- 

 nary plane as the real one, and vice versa. The figure will il- 

 lustrate this : C A X is the axis of x, C Y the axis of y, and 

 C K a line perpendicular to the plane of xy, or what may be 

 called the axis of imaginaries : C K and C L are the asymp- 



* Communicated by the Author. 



