322 Dr. Hirst on Equally Attracting Bodies. 



It is at once evident that all curves whose elements attract the 

 pole in the same manner as the corresponding elements of the 

 given curve must necessarily be situated upon the surface of this 

 cone ; and further, that if the latter, upon whose surface all the 

 equally attracting curves are supposed to be described, be un- 

 folded into a plane, a system of equally attracting plane curves 

 will result. On the other hand, too, it is just as evident that if 

 a plane containing any system of equally attracting curves be 

 folded in any manner into a cone, so that the pole may be the 

 vertex, a system of equally attracting curves, in general of double 

 curvature, will be obtained. 



For example, let the given attracting curve be a circle with 

 radius a, and let the attracted point be in the line through its 

 centre perpendicular to its plane. The equally attracting curves 

 will here be situated upon the surface of the right cone whose 

 vertex is the given point, and directrix the given circle ; let its 

 semi-vertical angle be a. If this cone be unfolded into a plane, 

 the circle with radius a will become an arc of a circle whose 



centre is the pole, the former vertex, and radius —. . The 



^ sm« 



angle subtended at the pole by this arc will be 27r sin «. The 

 plane curves whose elements attract the pole in the same manner 

 as the corresponding ones of this circular arc consist, as we know, 

 of the right lines which touch the circle to which it belongs. If 

 we conceive all these right lines drawn, and the plane sector con- 

 taining them refolded into the same right cone, they will all 

 become curves of double curvature, and will constitute the only 

 system of such curves whose elements attract the pole in the 

 required manner. 



Instead of examining more closely the general properties of 

 these curves, we will restrict ourselves to a still more particular 

 case. We will suppose the attracted point to be at the distance 



a \^'6 from the plane of the attracting circle. Then as «=^j 



sin a = I, the plane circular arc becomes a semicircle; and the 

 only line which attracts the centre in the same manner as the 

 whole semicircle is the tangent which is parallel to its diameter. 

 The centre being the pole, and the radius to the point of contact 

 our prime radius, the equation of this tangent line is evidently 

 rcos^=2« (24) 



Let the semicircle b(; now refolded into its cone, and let us seek 

 the projection of the curve, into which our right line has been 

 thus transformed, upon the plane through the cone's axis and 

 the generating line which was formerly the prime radius. Re- 

 taining the same point and line as pole and prime radius, we 



