38 KARL PEARSON 



We can accordingly repeat the process as often as we like and we have : 



M h,k)Jj^^ Q„Q„... Q^e-i""^' (Ivii), 



where t 2 = of + of + of + . . . + <rj, 



and after the operations have been performed we are to put all the cr's equal 

 to cr or %* = mo 3 . But no operator Q t affects any cr 2 in any other operator, and 



d d d s d s 



d^ = d% 2 ' TllUS (*'*)' d(<r 2 ) s may be put = ^ d ($ 2 Y ' and this makes a11 the 

 operators identical in form and we may write 



+ ... + (-l)V„(„')-j|^,+ . 



+ ...+(-l)'JV„( < r=) , j^y,+ .... 

 In this form of the operator we can now write .at once cr 2 = — % 2 and call 



1 nm 



m 



the expression Q t m . 



Thus Q- = l+N , { SJ^-N.(Sy^ + N,(%-y^ 



+ ...+(-l)-Ar„(Stj|i F + (Mi), 



1 TIT m *T m 



where N t = - 2 v„ N 6 = — v„ 



^ _ my, + ^m (to — 1) v* „ _ my M + m(m — 1) v 4 y s 



These values of the iV's rapidly converge and their values are given in Table III. 

 on p. 21 of this paper with those of the v's for a few values of n and m. As 

 we have seen on p. 32, they are the v's obtained by using values of nm for n. 



We now have the general solution of distribution from a centre : 



= (fiA^-WoL (A, + iV 4 Xl 4 + iV,n, + . . . + iV M fl M + . . . ) . . . (lx) . 

 This is absolutely identical with (xii), except that the constants v are replaced 



