124 SEE— DYNAMICAL THEORY [April 19, 



of the bodies of the system, it sufifices for us to deal with the integral 

 of (14). This triple integral admits of transformation by Green's 

 theorem (" Essay on the Application of Mathematics to Electricity 

 and ]\Iagnetism," Nottingham, 1828). If U and V be functions of 

 X, y, c, the rectangular coordinates of a point ; then provided U and 

 V are finite and continuous for all points zvithin a given closed sur- 

 face S, it is easy to show (cf. Williamson's " Integral Calculus," 7th 

 edition, 1896, p. 328; Riemann, " Schwere, Electricitat und Mag- 

 netismus," p. 73 ; Thomson and Tait's " Natural Philosophy," Part 

 I., Vol., I., p. 167; Bertrand, " Calcul Integral," p. 480): 



d[/dV dUdV dUdV 

 dx ~dy dy dz "ds 



dv r r r fd-v d^v d^v 



r r Bv rrr /b'v b'v b'v\ 

 = // 4^^ - J J J ^-l^ I? + ay + a? j ''^^yd.. 



The case in which one of the functions, U for example, becomes 

 infinite within the surface 6" was also considered by Green, and is of 

 prime importance in the present investigation of the theory of globu- 

 lar clusters. To simplify the treatment, suppose U to become in- 

 finite at one point P only ; then infinitely near this point U may be 

 taken as sensibly equal to i/r, where r is the distance from P. 

 Imagine an infinitely small sphere, of radius a, described about P 

 as a center. Equation (15) obviously is applicable to all points 

 exterior to this little sphere. Moreover, since 



/ ^2 a- ^5- \ I 



(,aP + a? + a70r=°' (^^) 



it is clear that the triple integral of the right members of (15) may 

 be supposed to extend through the entire enclosed space S, since the 

 part arising from the points within this little sphere is a small quan- 

 tity of the same order as a'-, and therefore of the second order of 

 small quantities. 



Moreover, since near P the function U is sensibly equal to i/r, 



