( 809 ) 



of rainshüwers lie between the asymmetrical limits : zero for the 

 smallest and oo for the largest values. 



Frequencies of this kind, therefore, offer an example of a transition 

 between the case of fixed limits and infinite limits on both sides. 

 As here there exists no symmetry in the limits, the zero-point cannot 

 be chosen so that, on integrating, the odd functions vanish, hence a 

 sepai-ation between even and odd functions would have no sense, 

 and ^\e are obliged to employ complete polynomia of ascending 

 degree. 



Here, as in the case discussed in § 2, there is no advantage in 

 making the origin of coordinates coincide with the arithmetical mean 

 and, from a logical as well as a practical standpoint, the zero-limit 

 is indicated. 



In order to develop the function between the limits oo and zero, 

 the only thing to do is to multiply the series of polynomia with a 

 suitable factor e.g. e~-'^ , so that the equation of the frequency curve 

 becomes : 



ic = e-^ (^0*^0 + ^1*^^ + . . . . etc.) 



= ^oV^o + ^1^1 + ete. 



where : 



Sn = .t'" -\- a^x -j- aj.i'"-2 -|- . . . . an. 



The conditions to be satisfied by the a coefF. are then : 

 j e-^ Sn dx = , I e-^ xS,, dx = . . . . I e^^ a-"— i aS„ dx = 







and as : 



ƒ 



e~-^ ,t'" dx z= nf, 

 

 the general conditional equations are : 



n f + (n — l) / a, + {n—2)f a, + 1 / a„_i -f / a„ =: 



(n + 1)/ + n!a, + (n-l) f a, + . . . . 2 / a„_i + 1 / a„ = 



(2n -1) / + (2n-2) / a, + (2n-3) ! a, -^ . . . . n ! a„_i + (n-1) / a„ = 

 Hence we find for the general expression : 



Sn = x^^ — - .t'"-i + .t'«-2 — {—l)n n ! . (16) 



J- . 0; 



The method of calculating An is the same as in the former cases 

 as here too : 



