342 Mr. Stubbs on a new Method in Geometry. 



circle passing through the pole ; in the circle, the centre being 

 the pole, this becomes a very remarkable theorem. 



In two inverse curves the differential elements of the arcs 

 are connected inthe following manner: — d s' : d s : : r' : r, or 



d s' = —5- d s ; hence the differential element of the arc of a 



curve can be known when that of its inverse is known. This is 

 remarkably connected with the theory of elliptic functions ; the 

 arc of an ellipse being represented by an elliptic function of the 

 second kind, the arc of the curve formed by the intersections 

 of the perpendiculars to the diameters of an ellipse through 

 their extremities, by one of the first kind, I have found by this 

 method that the arc of an inverse central ellipse is represented 

 by A II (w, <J>) — B . F . <p, A and B being constants, and II being 

 an elliptic function of third kind with a circular parameter*, 

 and F <p one of first kind ; the amplitude <J> being the angle by 

 which the amplitude of the arc is measured in the ellipse from 

 which it is generated ; the x of the corresponding point of el- 

 lipse being = a sin <£, y — b cos <p. Hence from the gene- 

 ral formula for the comparison of elliptic functions of the third 

 kind (since if cos <r = cos <p cos \|/ — sin <$ sin ty v" 1 — c 2 sin 2 <r 

 F <J> + F v(/ — Ftr = 0), viz. 



tt/ ^ , tty i\ nf \ 1 * _i rc 4/ a sin \J/ sin <J> sin <r 

 U(n<p) + II(w^)— n(w<r) =— -^tan- 1 — , A , 



* ' K v */ a 1 + n— wcos\l/cos<f>cos<r 



an infinite number of arcs of an inverse central ellipse may be 



found such that the difference between one measured from the 



vertex and the other between two other points shall be equal 



to a circular arc; and if the difference of two arcs of an ellipse 



be equal to a right line, the difference of the arcs inverse to 



these shall be equal to a circular arc. 



I will not trespass on your limits by proving this, it may 



be shown by the ordinary method ; I will merely state 



a 2 — Z» 2 

 the values of the constants \ n the parameter = — ^ — ' 



A = \W + nb* )' * ~n~W' a and h bein S the axes 

 of the common ellipse, c its eccentricity, Jc the modulus of the 

 inverse curve, or the constant equal to the rectangle under the 

 coincident radii of the two curves. By discussing the formula 

 above given for the comparison of elliptic functions of the 



third species, substituting for n and a (ais(l+n) (1-| ) ), 



* This is not as general as I could wish, as there is a relation between 

 the modulus and parameter which properly should be independent. 



