26 KAEL PEAESON 



Transfer the differentiations from /3 to o- 2 and we have : 



= (- 1 ^)^ ^ 



or, all the w-functions can be found by differentiating the first w-function with regard 

 to the standard-deviation squared. Then by (xii) we have 



(xlvi). 



Thus, if we put cr 2 — t : 



But: \\M+f)*y=^\y^*y^-^l^* 



V27TO- / 



Now I e t l<T dc = (T \ e * x dx, and this integral vanishes when c = oo . Hence 



J 00 J 00 



^ (c) =^( i+ ^^-- i w + - + (- i )-""'-^ + )/y i ^ 



(xlvii). 



Since .F n (c) clearly vanishes for c infinite, it is not needful to introduce a 

 constant. 



It remains accordingly to determine the successive differentials of the integral 

 with regard to t. Call the integral i; then, if yj^c/cr — c/Jt, 



di -W d-n \ G - C 2/2t c . . 



dt=- e dt=2$ e =»-*. = «W^ 



By (xlv) we know that d i '(i) /dt s = ( — l) s cii M lt s . Hence differentiating s — 1 times 

 we have : 



d s i TTG/d s - 1 a,_ . d'-'cup 1 (s - l) (s - 2) d'~'o) B 1.3 1 \ 



- 1 ( s V d?- 2t + 2! rfrOf" J 



(-l)'-' gc/ ,, ^ (s-l)(s- 2) 1.3 



^ ^«.(.-w + (« ~ !) w 2( S - 2) i + j-^ <" 2(g _ s) — 



( s -l)(s-2)(s-S) 1.3.5 \ 

 1.2.3 2(s_4) 2 3 j ' 



cft g V« V d£ s 



