Goodness of Fit of Frequency Distributions. 373 



Now* 



P _ ^x 



I e-^x<t- 2 dx 

 Jx 



i e-^\^d X 

 Jo 



= ~ 







-e- h x\«-*d x 



I e-*x 2 x <i~2dx 



Jo 



or e -hX 2 Y q-z 



J o « _ix x*~ 2 <*x 



= iV Xl Wx)> 

 where . e -ix- v i-3 



^X) = 7^ * 



Now suppose ^ is even : then 



J o e-ix\«-*d x = (q-3)(g-5) ... 1 x J V^ X . 

 Thus, since 



J»oo 

 o 



w=\Zf*-*' V2.iT.~ 3 .g-3 



= P ? (X 2 )-P,- 2 (% 2 )t. 

 Next suppose q is odd : then 



f ,-ix 2 %5 "% = (?-3)( ? -5) . . . 2 f •-*x^i x . 



Jo Jo 



Thus, since 



I <r** 2 x*K = 1, 



f'(x)=P,(x 2 )-P 9 -a(x 2 )t, 

 the same value as before. 



* Phil. Mag. loc. cit. p. J 58, Eq. (iii.), remembering that n=q — l 

 — number of independent variables, 

 t Phil. Mag. loc. cit. p. 159, Equations (v.) and (vi.). 



