166 Dr. Hirst on Equally Attracting Bodies. 



sections of a cone whose vertex is the centre of inversion. 

 When the plane of one circle passes through this centre of inver- 

 sion O, it coincides, of course, with the plane of the other circle, 

 and when one circle passes through 0, its inverse is a line in 

 accordance with the converse of III. 



VI. The inverse of a surface of the nth order is, in general, a 

 surface of the 2/ith order. 



For in general every circle through the point of inversion will 

 cut the given surface in 2n points, and hence by III. every 

 right line will cut the inverse surface in 2n points. Similarly : 



VII. The inverse of any plane curve of the nth order, the cen- 

 tre of inversion being in the same plane, is a curve of the 2rath 

 order. 



Since the infinitely distant line cuts the original curve in n 

 real or imaginary points, it is evident from III. that the inverse 

 curve will have a multiple point (real or imaginary) of the nth 

 order in the centre of inversion, and besides this every multiple 

 point and cusp of the original curve will have its corresponding 

 one of the same order in the inverse curve, and so on. 



VIII. The inverse of any surface is at the same time the locus 

 of the foot of the perpendicular let fall from the centre of inver- 

 sion upon a tangent plane to the reciprocal polar of the given 

 surface with respect to the sphere of inversion. 



For, as is well known, the polar plane of any point cuts the 

 radius vector perpendicularly in the inverse of that point. 



IX. The inverse of a line of curvature on a given surface is 

 also a line of curvature on the inverse surface. 



For the normals in two consecutive points m and m^ of a line 

 of curvature on the given surface meet in a point fi, so that if, in 

 the planes Ofjum, 0/im„ two circles be drawn so as to pass 

 through O and touch the normals, respectively, in m and m^, 

 these circles will not only cut the surface perpendicularly, but 

 they will intersect again in a point fju^, of the line Oyti, such that 



But by III. the right lines, inverse to these two circles, are the 

 normals to the inverse surface in the consecutive points m', m\ 

 corresponding to m, m„ and since these normals meet in a point 

 /u,'i, inverse to /i,„ m' m\ must be an element of a line of curva- 

 ture of the inverse surface. 



From the above, we conclude, too, that the principal centres 

 of curvature ^ and ix\, at corresponding points of inverse sur- 

 faces, correspond, but not as inverse points ; they are, in fact, 

 connected by the relation 



