Researclies into the theory ot pr(jhahiHty. 25 



|2I?, = i>'„(X^-(2X+l)v/ + v;), 

 ^ ' \3B, = B, {V - (3X^ + 3X + 2) v/ + 3 (À + 1) v,' - v,'), 



|4 i', = i?, (X* — (4X3 ^ (3X2 _^ gx ^ (•) ^ ^ßX2 ^ +11) v.; 

 -(4X + 6)V3' + v/), 



We give to X such a value that the coefficient vanishes. We then have, 

 putting v/ = i, 



5, 0, 



|2i?, = i?,(v; -&2-&), 



|3 i?3 = j5, (— 2W — 36^ - 2?> + 3?;v,' + ?>v: — V3'), 

 |4 B^ = B,, (— 36* — 86^ — 6& + {61/ + 126 + 1 1) v^' 



-(46+G) v; + v;), 



We here introduce the moments about the mean tliat are defined by tlic 

 equations 



(17*) \>-,^^. = :ù(x- hyF(.,) [s = 0, 1, 2, . . .), 



h being the coordinate of the n^ean, so tliat 



V3' = V3 + 36v, + //■', 



v; = V, + 46V3 + <:;6^, + 6^ 



which relations are obvious, if we remember that the mean is determined in such 

 a manner that the first moment about it vnnishes. 



The expressions for i?^, B.^ and J?^ now assume the simple form 



12i?, = I?,(v,-6), 

 (17) |3i?3 = 2;„(-V3 + 3v,-26), 



|4 B^ = I?, (v, — 6V3 ~ G6v, + 1 1 + 36^ — 06), 



When the moments about the mean are known, the coefficients i?.,, B.^, B^ 

 are easily obtained from (17), and we have 



(17**) F{x) = ij.,fj,r) 4- B,^''\ + B,t^^^\ + B^à''\ + . . ., 



where now X = 6 = v/. 



2:o. Wc put co=l, leaving c undetermined. 



If we employ the parameters c and X to make vanish the coefficients B^ and 

 B^, we now have 



Luuds Uiiiv:.s Årsskrift. N. P. Afd. 2. Bd 1. '4 



