1912.] OF THE GLOBULAR CLUSTERS. 123 



with similar expressions for 



dF dV dV 



dx' dj' dz' 



The integration for the mutual potential energy of the stars in 

 the outer shell relative to those in the central sphere of radius r 

 leads to a sextuple integral 



C C C C C C (t<t' dxdydsdx' dy' dz' 



^ J J J J J J V{x' ^-^xf + {y' '^f~j^{z' - zf ^^^^ 



And the total of the mutual attractive forces resolved along the 

 coordinate axes are 



an C C C C r C <^°"'(^' — 'r)dxdydzdx'dydz' 



^^^ J J J J J J V{x' - xf + {y' -yf + {z' - zf-]V ^'^^ 



with similar expressions for 



an an 



dy ^ dz' 



Now it is easy to prove (cf. Thomson and Tait's "Natural Phi- 

 losophy," §§ 547-548) that the sextuple integral (11) can be put into 

 the form 



n = f j f(^ Udxdydz = f f f<^' Vdx'dy'dz'. ( 1 3) 



By actual derivation of the expressions (9) we easily find that 



dU dV dUdV dUdV 

 dx dx dy dy dz ~dz 



is equivalent to fi, by (11), and therefore 



rrrfdUdV dUdV dUdV\^^^ ^ ^ ^ 



}jj\-dx^r + -d-y-d-y-^^z'd-z)^'''^^"^'-'^'^^' (^4) 



47r being introduced owing to the integration over the closed sphere 

 surface (cf. Williamson's " Integral Calculus," edition of 1896, p. 

 330; Bertrand, " Calcul Integral," p. 480). 



As the right members of (13) give the mutual potential energy 



