Mr. Stubbs on a new Method in Geometry. 34*3 



when the arcs are measured from the extremities of the axes, 

 I come to the following theorem. 



The difference of the arcs of an inverse ellipse, one mea- 

 sured from the end of the major, the other of the minor axis, 

 and whose amplitudes fulfil the condition b tan <$> tan \f/ = a 

 {a and b being the axes of the direct ellipse), is equal to the 

 arc of a circle, which may be found by the following construc- 

 tion : — let I be the intercept between the foot of the perpendi- 

 cular from the centre on the tangent of the direct ellipse and 

 the point of contact, P the perpendicular from the centre on the 

 line joining the extremities of the axes of the direct ellipse, andL 

 the line joining the extremities of the axes of the inverse ellipse; 

 then taking a line = P, raising at its extremity a perpendicu- 

 lar = I, and producing the line P v n 

 until the whole line produced = L, 

 with the common extremity O as 

 centre, and L as radius, describe 

 a circle, the arc of this circle in- 

 tercepted between the other extre- 

 mity, M, and the line joining O with the end of the perpendi- 

 cular I, is the difference of the required arcs. The analogy of 

 this theorem to Fagnani's with regard to the direct ellipse (by 

 which the difference of the corresponding arcs of it is I) is 

 obvious. The area of the inverse ellipse is an arithmetic mean 

 between the areas of the circles described on its axes. 



To apply the inverse method to surfaces I will state the fol- 

 lowing principles : if one surface be inverse to another, a tan- 

 gent plane being drawn at one point, the tangent plane at the 

 inverse point is had by bisecting the line joining the points 

 by a plane perpendicular to this line, and through the line 

 where it cuts the tangent plane to the first surface, and the 

 inverse point we draw a plane ; it is a tangent plane at the 

 inverse point: this is readily seen, as if through the common 

 radius we draw any plane cutting the surfaces in two curves, 

 these curves are inverse, and the construction which I gave for 

 the tangents at inverse points makes this construction evident. 

 Hence the normals at inverse points of surfaces are in the 

 same plane and equally inclined to the common radius. 



From this construction for the tangent plane, it follows that 

 if two surfaces cut at right angles their inverse surfaces cut at 

 right angles. Hence if we describe the developable surfaces 

 formed by the tangent planes and normals at the points of a 

 line of curvature, since these surfaces cut at right angles their 

 inverse surfaces cut at right angles at the inverse points of the 

 line of curvature, but the surface formed by the tangent planes 

 to the inverse surface at those points is touched by the inverse 



