164 Dr. Hirst on Equally Attracting Bodies. 



then, by hypothesis, 



Om . Om' = 0/K, . Ow'i = const., 



so that the triangles mOniy, nJOm\, which have in common an 

 infinitesimal angle at O, are similar, and the angles m^mO and 

 m'm',0 equal. Ultimately, thei'efore, the angles myinO and 

 m\m'0 will be supplemental, in other words, the corresponding 

 linear elements mw,, wW, are equally inclined to their common 

 radius vector Om. Since the same is true for all other corre- 

 sponding linear elements around m and m', it follows at once 

 that the plane of all the former, or the superficial element at m, is 

 inclined towards the radius vector at the same angle as is the 

 plane of all the latter, and consequently that the corresponding 

 superficial elements atti-act equally. 



7. Inasmuch as inverse surfaces will simultaneously present 

 themselves in all our subsequent solutions, it will be worth while 

 to state briefly some of their principal properties. 



Ai'ound a fixed point (the centre of inversion) , and with any 

 radius c, conceive a sphere to be described ; then any two points 

 m and m', situated on the same line through O, and on the same 

 side of the latter point, are said to be inverse to each other when 



Om . Om' = c^. 



This being premised, inverse figures or systems of inverse points 

 possess, amongst others, the following properties : — 



T. Any two lines whatever (right or curved) cut each other 

 at the same angle as do their inverse lines. 



For if TO???j, mm^ be the elements of one pair of lines at their 

 point of intersection m, and m'm\, m'm'^ the corresponding ele- 

 ments of the inverse lines, it is evident from art. 6 that the pair 

 of corresponding elements mm,, m'm\, if produced, would meet in 

 a point jjby and form an isosceles triangle on mm' as base. Simi- 

 larly, mm^ and m'm'c^, would if produced, meet in a point ijl^ and 

 form another isosceles triangle on the same base. This being 

 the case, it is easily seen that the two triangles /M^m/j,^, fi^m'/j,^ 

 have all their sides equal, whence results the equality of the an- 

 gles fi^m/j,^, fifm'fMc^ as asserted in the theorem. 



Amongst other consequences of I. it is evident that the inverse 

 of the normal to a surface cuts the inverse surface perpendicu- 

 larly. 



II. The inverse of a plane is a sphere through the centre of 

 inversion, whose centre is on the perpendicular let fall from the 

 centre of inversion upon the plane. 



For if p be the foot of this perpendicular and p' its inverse, m 

 any point of the plane and m' its inverse, we conclude, as in art. 

 6, that the triangles mpO and p'tn'O arc similar. The angle 



