â4 c. V. L. Charlier. 



It is now 



X = — 00 



in which formula \j must assume all values given by the relation 



y = xiS) -\- c, 



where a? = 0, +1, +2, +3, ... in inf. 



As to T,\dc) we know that it is a polynôme of degree r in x. \i we write 



T,\x) - o^a;'- + S<;)ic'-' + ■ • • + ^^V-.x + §<'', 



and ohserve that 



l-t/' = CO ^ {]) — c)' F{y) = X [rw]' F[c -\- vm) 



is dependent on c, but independent of to (if lo is rather small), we have 

 (o' ^i?,. = S'o'ix," + 5^>co[j.,„/' + . . . + 5S.'L,co'-'ix/' + 8ir'co>;', 

 so that the values of B^^, B^, . . . now are 









0^'B, 



= Xo)[j.j," - 



- [J'l"' 



' !2 B, 



= X\oVo" 



-(2X+ l)co|j./' + |x,", 



^l'àB, 



= X\oVo" 



_ (3X^ + 3X + 2) co^ix/' + 3 (X + 1) ..[x," - [x,", 







— (4X-^ + GX^ + 8X + 6) toVi" + (<3>^' + 12X + 11) o^^^x^' 



— (4X + 6) wix/' + [x/', 



The frequency curves of the type B may be treated mathematically in diffe- 

 rent manners. In the general formula (13) oj, b and X may be arbitrarily chosen. 

 The greatest convergency is generally attained if these constants are determined 

 in such a manner that Bj^ = B^ = B^ = 0. It is, however, not necessary to choose 

 the parameters in this manner. Sometimes it will be found convenient to give to 

 X, c or oj determinate values. We will treat some of these values. 



l:o. We put CO = 1 and c = 0. 



It is now 



(15) F{x) ^ B,^^(x) -{-B,^'!^ + B,^^-^B,^^^... 



Dividing the expressions for B^, B^, B.^, ... by B^, we obtain, if we put 

 (16*) !J'o'v/ = ^'' 



