jftolotropic Potential. -4G5 



form of statement, in which density p c is so defined that 

 -±7rp occurs in the flux equation, and make*/o =l. For 

 isotropic potential we take the area a of each section of the 

 body by a plane passing through the point at which potential 

 is reckoned : it is a function of the angular position of the 

 normal N to the plane. It' dco s is an element of area about 

 the position of X on a sphere of unit radius, the statement 

 i- that \ ad<o s is 27r times the potential, the integration 

 covering the sphere. 



When the attracted point is taken as the centre of a 

 polar system, the element of area is h' 2 d<f>, that of potential 

 is \r-Jw v {cf. §6) : there may be one section r 2 or >y — >'i~, 

 or several sections according- to the position of and the 

 shape of the body, but the integrand is alike in the two cases. 

 The first integral has three angular integrations, one in the 

 plane and two comprised in dco y , and the mode of integration 

 must be altered so as to bring together the elements which 

 are plane-sections of the cone about OP. Thus N makes a 

 circuit about P, while the plane element to which it is 

 normal is a section of the cone through its axis in various 

 azimuths, and we may denote by % an angle defining the 

 position of X in the circuit. When P is taken as pole 



/\ . /\ 



da M is represented by dyd . PN, since PJN differs by an 



infinitesimal from a right-angle ; and in a like manner d<o P 



/\ 

 i- represented by dcfrd.PN. Thus dm s and dco P have a 



/\ 

 common element d .PN, the angle d<j) of the plane section is 



th<m used to complete the specification of d(o P , with the 

 result that d^dco P represents the element of the triple integral. 

 The position of X in the circuit about P is indifferent, 

 because the various plane sections of the cone are not distin- 

 guished : thus the integrand does not depend on ^, and the 

 circuit integral is 'l7rdco P . 



The s] tecial cases so far verified are : — The frustum of a 

 cone for the vertex, sphere for all points, spheroid for 

 external points on the axis of revolution, and a ring of finite 

 circular section for all points on the axis of symmetry. The 

 ring is the only non-quadric surface which has been 

 examined ; the work for the cone elucidates the general 

 proof. 



The theorem appears to be a general theorem of integration 

 on a sphere. Thus if /(Pj is a function of position on a 



sphere, and in mean for a great circle, pole N, viz. \/(P)dcf) 



