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XXII. — Investigation of Analogous Py^operties of Co-ordinates of Elliptic and Hyper- 

 bolic Secto7's. By William Wallace, LL.D., F.R.S.E., F.R.A.S., M. Cambridge 

 P.S., Sfc. Emeritus Professor of Mathematics in the University of Edinburgh. 



Read 15th April 1839. 



The analogy between certain properties of the co-ordinates of elliptic and 

 h3rperbolic sectors, which forms the subject of this memoir, was observed by the 

 early writers on the fluxional or differential calculus, and employed by them in 

 its improvement. But, independently of this important apphcation, the proposi- 

 tions in question are some of the most elegant theorems in geometry, and highly 

 interesting as abstract truths. 



Maclaurin, in the third chapter of the second book of his Treatise of Fluxions, 

 has proved the truth of this theorem : " Supposing 7i to be any number, let E and 

 w X E be elliptic sectors, which stand on arcs that begin at a vertex of either 

 axis, and H and nxH hyperbolic sectors, which stand on arcs that begin at the 

 extremity of the transverse axis : the algebraic equation which expresses the re- 

 lation between x and z, ordinates drawn to the other axis from the extremities 

 of the arcs, must be the very same in the two curves." 



From this he has inferred, that if the relation between the ordinates x and z 

 of the one curve be found, it may be assumed as equally true of the ordinates of 

 the other curve, and thus the properties of the one may be deduced from those of 

 the other. 



The late Professor Playfair, in a paper in the London Philosophical Trans- 

 actions for 1779, has, by the use of the symbol of imaginary quantity (viz. V — \) 

 deduced from the geometrical properties of the circle, which are the foundation of 

 the angular calculus, corresponding properties of the hyperbola ; but I do not 

 remember any writer who has, by a single process of legitimate geometrical or 

 analj^tical reasoning, established the identity of the properties of the two curves ; 

 although such an investigation would have been a fine example of the power of 

 modern algebraic geometry as an instrument of research, and a valuable addition 



* 



to treatises on the conic sections constructed by that compendious and elegant 

 mode of reasoning. 



2. Considering it desirable that a subject so purely elementary should be brought 

 under the dominion of the ordinary doctrines of analysis, I have endeavoured to 

 include the properties of both curves in one set of formulae ; taking from geometry 

 the very least assistance possible ; and avoiding all use of the imaginary symbol 



vol. XVI. PART II. 3 s 



