346 Mr. Stubbs on a new Method in Geometry. 



perties of curves of the second order, we come to those of the 

 fourth with regard to polar circles, so by deducing those of 

 the second order with regard to these latter, we might arrive 

 at the properties of the higher curves with regard to lines. 

 I shall not trespass on your limits any further by noticing 

 new properties, many of which I have deduced in the paper 

 before alluded to. I shall merely show the application of this 

 method to physical investigation by two simple instances. 



1. Since the ultimate elements of two inverse surfaces cor- 

 responding to each other are inversely placed 



with regard to the common radius, by de- Fig. 6. 



scribing an infinitesimal cone having these ele- 



ments for their bases, —^ = -^ , m and m' 



being the masses of the bases, and dand d' the 

 distances : hence the attraction of two infini- 

 tesimal elements resolved in any direction 

 are the same, or the whole attractions of the corresponding 

 parts of two inverse surfaces on the common pole is the same. 

 The application of this to the plane and sphere is obvious. 



2. The second theorem I shall state regards the wave-theory 

 of light. It is stated by Sir John Herschel in his Essay on 

 Light, that the equation for determining the velocity of a 

 wave perpendicular to a given plane z = m x + ny, is 



(V2 - a*) (V 2 - £ 2 ) + m* (V 2 - 6 2 ) (V 2 - c 9 ) + n* ( V 2 - a 2 ) 



and that this equation is had by an elimination which he states 

 to be very complicated: it can be had at once by the following 

 geometrical method. 



Taking with him the equation of the surface of elasticity to be 

 R 4 = a 2 .r 2 + i 2 ^ 2 + c 2 z 2 ; 

 if we find the intersection of this with the concentric sphere, 

 we get 



■r 2 (a 2 -r 2 ) + y*(b*-r*) + s 2 (c 9 -r 2 ) = 0; 



but if we put r = to a constant, this represents a cone of 

 second order. Now if we draw a tangent plane to this cone 

 through the centre, the line of contact must be the axis of the 

 curve cut out as the tangent to the sphere whose radius at r is 

 perpendicular to it at its extremity; but this is also contained 

 in the tangent plane, and therefore the intersection of these 

 two planes is a tangent to the section perpendicular to its dia- 

 meter, which is therefore an axis, but identifying the equa- 

 tion of the tangent plane to the cone, viz. x x l (a 2 —r 2 ) + yy 1 

 (6 2 — r 2 ) + zz' (c 2 — r 2 )= 0, with the plane Ix + my + nz = 0, 

 I ;= x 1 (a 2 — r 2 ) m =s i/ (6 2 — r 2 ), 8cc, x' y' z' being the coor- 



