( 8J2 ) 



we find the following relations : 



d" , , f/"+' dxbn 



tf'.+l = ( -1)" — (e--- .t"'+') = ( - 1)"+' .f -y~TT (^--^ 't'") = - .f -f- 



.t- + (2 - .v) " + n7"„ = 



ffr'^ fir 



dx-" dx ^ ^ ^ ^ ^ 



^ T, + ^ -V— + (^ + ^) ^"+1 = ^• 



dx^ dx 



In exactly the same manner as the i?-senes could be expressed 

 in diff. quot. of the Q series : 



till z=z{x' — l):^ An — -— , 

 dx 



SO the ip' series might be expressed in diff. quot. of the tp series : 



u.i' ■=. — X ^ A,i . . 



dx 



In dealing with this kind of frequency curves an alteration of the 

 scale value offers great advantages as well as in the case of fixed 

 limits. 



In the case discussed in § 2 it was possible by this artifice to 

 simplify the limits; here such an alteration has no influence upon 

 the limits which remain and oo if we write hx for x, but we are 

 able by this means to accomodate the first term of the series, by 

 which the area is determined, according to the form of the curve, 

 so that the task of the ^-coefficients is lightened. 



By the factor h, which by its nature is a positive quantity, no 

 complication in the calculation of the constants is introduced : the 

 series is now : 



u — e-f'-' [A,S,{hx) -f A,S,{hx) + . . . . etc.]. . . (23) 

 and, because: 



J I 00 -I ^ CO 



e-^^ Sn{hx) Snihx) dx = - I <?-' S,,{t) S„{t) dt 







\ji!n! if ?if{n—l)f^ 



(-I);;- 

 nf 



(24) 



We might also omit the coeff. h in (24) and write (23): 



u — he-''^- [A,S^{hx) -{- A,S,{hx) -{- .... etc.] . . (23«) 



