( 813 ) 



The scale value h may, of course, be chosen quite arbitrarily ; 

 it is however desirable to do this in accordance with the nature of 

 the curve and, therefore, to calculate it methodically from the y(\\Q\\ data. 



This can be done by suppressing- one of the vl-conslants in (20^/) 

 so that the mean value corresponding with this constant can be 

 made use of to define It. 



If then we put : 



^, = 



we find, as A^ = 1, 



§ 4. Development between the indefinite limits ±: oo . 



Hy reasons of symmetry, in this case it is logical to take (?-•'- for 

 the factor by which the limits are determined and when, for the 

 same reason, the arithmetical tnean is chosen as the oj-igin, the poly- 

 iiomium can, as in the case of tixed limits, l)e separated into even 

 and odd functions, because then, on integrating between the limits, 

 the odd functions vanish. 



The series becomes then : 



u ^ e-^"- \_A,U, + .1/^, ^A,U,-^.... etc. 

 = A,ip, + A^<p^ 4- A,(p^ 4- . . . . etc. 



as, by the choice of the origin, the Jj-term has to be omitted. 

 The conditional oipiation for the determination of the a constants is : 



J- 



,^•"' (/)„ ^6' = nc<^n (25) 



or. generally, for n even: 



|;(,,_l)(„_3)...l] .|_2aJ(/.-3)(n-5)...ll f 2^ [(« - 5) (;?-7)...l] + 



+ - -f- 2"/2a„==0 

 [(;i4-l){«-l)...l] + 2a, 1(m_1)(,,_8) . IJ + 2^, |(n-;J)(M-5)...l] 4- 



+ •• + 2"-2/2a„ = 



[(2n-3)(2«— 5)...ll + 2a, [(2n-5) (2n-7)...lJ -f 



+ 2»a,, [(2/1-7) (2«-9)...l] + ... -f 2«„ [(n-3) (/» -5)...l] =: 

 and, for ii odd : 



55 

 Proceedings Royal Acad. Amsteidam. Vol. X. 



