Dr. Hirst on Equally Attracting Bodies. 1 67 



Analogous theorems with respect to inverse curves, of single 

 and double curvature, might easily be established ; to do so, 

 however, would be to extend too much the present digression. 

 We return, therefore, to the subject under consideration*. 



8. A sphere around the attracted point as centre is but an 

 individual of a large class of surfaces which possess peculiar 

 attracting properties. Any surface of this class may be defined 

 as cutting eveiy ray of the pencil whose centre is the attracted 

 point at the same angle /S. The densities at corresponding 

 points being everywhere assumed to be the same, it is evident 

 that the class referred to includes innumerable groups of equally 

 attracting surfaces, each group corresponding to a particular 

 value of /3. The sphere belongs to the group determined 



by the value y3= jr. The attractions of con-esponding portions 



of two surfaces belonging to different groups (/3) and (/3,) 

 are like-directed and have a constant ratio to each other, that 

 is to say, a ratio which is independent of the magnitude of the 

 attracting portions. This is at once evident from the equa- 

 tion (1), which shows that the attractions of corresponding ele- 

 ments are inversely proportional to the sines of the constant 

 angles /S and /9j. As a particular case, therefore, it follows that 

 the attraction of a portion of a surface of the group (yS) is to 

 that of the corresponding portion of the sphere as 1 : sin /S. 

 Hence, as far as attraction is concerned, surfaces of this class 

 enjoy all the symmetrical properties of spheres. For example, 

 if the density on such a surface be everywhere the same, any 

 portion of the surface intercepted by a cone, having an axis of 

 symmetry and its vertex at the attracted point, will attract this 

 point along that axis, and so on. 



The inverse surfaces of those here considered clearly belong to 

 the same class and group ; the general equation of the class will 

 be given in the sequel ; in the mean time, however, it is easy to 

 find individuals ; such, for instance, would be the surface gene- 

 rated by the rotation around any radius vector of a plane loga- 

 rithmic spiral, having its pole at the attracted point. Or, more 



* The subject of inverse figures appears to have beeu first introduced 

 under the name of " electrical images " by Prof. W. Thomson in the Cam- 

 bridge and Dublin Mathematical Journal (vols. viii. and ix.); and subse- 

 quently, under the title " principe des rayons vecteurs r^ciproques," it 

 received considerable development from M. Liouville. It was not until the 

 present memoir had been written that I discovered how closely the above 

 art. 7 coincides with a portion of the excellent memoir of M. Liouville 

 (Journal de Math. vol. xii.). Notwithstanding this coincidence, however, 

 I have allowed the article in question to stand, as well in order to avoid 

 the inconvenience of reference, as on account of the demonstrations there 

 briefly indicated, which differ somewhat from those given by M. Liouville. 



