274 Dr. Hirst on Equally Attracting Bodies. 



We may remark, once for all, that in the integration here indi- 

 cated it will not be necessary to add an arbitrary constant, snice 

 the same may be regarded as already involved in u (sec equa- 

 tion (5), art. 11). 



The geometrical peculiarity of the above solution is, that any 

 cone of revolution about the a?-axis, having the origin for vertex, 

 cuts the surface in a curve which is everywhere equally inclined 

 to the generators of the cone, so that on unfolding the latter into 

 a plane, the curve in question would become a logarithmic spiral 

 having the origin for pole. 



It will be readily seen, too, that the complete integral of (25) 

 will result from the elimination of u between the equations 

 w = F(^, «)+«(/.+/(«), 1 



where f{a.) is an arbitrary function of a, and, as usual, 



/'(«)= ^i^. 

 •^ ^ ^ da 



In fact, if we differentiate the first of these equations under the 

 hypothesis that a is a function of and (j> as determined by the 

 second, we shall, in consequence of this second, re-obtain the 

 values (26), which we know to satisfy the equation (25). 



According to the method (Monge's) here employed, eacb of 

 the required surfaces is considered as the envelope of the space 

 traversed by the first of the surfaces (28) when the latter changes 

 its form and position in consequence of the variation of the para- 

 meter u. When u is regarded as having a definite, though arbi- 

 trary value, the two equations (28) represent the so-called cha- 

 racteristic of the required surfaces, — a curve of double curvature 

 which may clearly be represented equally well by many other 

 pairs of equations, and amongst the rest by the pair 



w=r(^,«t)-«^+/(«)-«/'(«), 



0=^ + <^+/(«), 



(29) 



of which the former is the equation of a surface of revolution 

 about the a?-axis, and the latter that of a cone having the origin 

 for vertex. 



30. By means of these equations (29), it is easy to verify a 

 remark already made in the first paragraph of art. 19, to the 



