262 



This singular result appears to the author likely to lead to some 

 new law of electricity, and tends to overturn the present ideas of 

 galvanic excitation. An investigation of the details is promised at 

 some future period. 



2. Investigation of analogous properties of Co-ordinates of 

 Elliptic and Hyperbolic Sectors. By W. Wallace, 

 LL.D., Emeritus Professor of Mathematics, University 

 of Edinburgh. 



Tlie object of this paper is to investigate, for the ellipse and 

 hyperbola, a series of propositions analogous to those which, in the 

 circle, constitute the calculus of sines. 



Any diameter of an ellipse, or a transverse diameter of a hyper- 

 bola, is expressed by a, its conjugate diameter by b, and the ratio 



of these diameters, viz. -, by c. The nature of the curves is ex- 



b 



pressed by this formula, x^ + cy^ = a^. In the ellipse, c has the 

 sign +, but in the hyperbola the sign 



Any two sectors of the curves contained between any diameter 

 of an ellipse, or a transverse diameter of a hyperbola, and any other 

 diameter, are denoted by a and /3, and sectors equal to their sum 

 or, difference hy ec-\- ^ and « — /3. The co-ordinates of the vertices 

 of the diameters which bound a sector », are denoted by/(«), and 

 F(«). These co-ordinates being considered as functions of the 

 sectors, the letters/and F are used as characteristics of the func- 

 tions; when applied to the circle, /(a) is the cosine of », and F(<«) 

 its sine. The expressions for / (« -f /3), /(« — /8), F (« -f- ^), 

 F (« — ,3), the origin of the sectors being any diameter of the 

 ellipse, are investigated, and the results tabulated. 



Supposing the sectors to be contained between a, the transverse 

 axis of an ellipse or hyperbola, and any oblique diameters, then 

 putting e for the eccentricity of the curve, the formulae found are 

 these : 



«/(« + /3)=/(.)/(/5)--j^,F(«)F(/3), (1) 



aF(« + /3) = F(«)/(^)-(-/(«)F(A), (2) 



s 



rt/(«_;S) = /(«)//3-h-^FWF(r.), (3) 



«F(«-,3) = F(«)/(/3)-/(«)F(/3) (4) 



Proceeding from these formulae, a theory of the functionsy'(wtt), 



