(2) uV = -^.U'V^. 



2 C. V. L. Charlier 



properly chosen, we then have between u and v on one side and U and V one 

 the other, the relations 



(1) U=ru; r=^rv, 



which give 



1 



1 



r , 



For computing the characteristics of the correlation surfaces we take recourse 

 to the method of moments. Regarding the relations between the moments and the 

 characterics I refer to Meddelanden N:o 4 and to the following paragraphs. The 

 moments may be taken either about the mean or about the origin. Let 



(f[u, v) 



be the correlation function of u and v, and 



the correlation function of the linear velocities U and V. Then the moments v'^ of 

 of the order i in n and j in v about the origin is given by the formulée 



(3) v' ^. = jj du dv f{u, v) mV 



— 00 



and the moments m.j about the mean [x^, by 



(3*) V =ffdudv f{u, v) {u — Xof {v — %y. 



— GO 



In like manner we have, for Ü and V, for the moments N\j about the origin 

 the expression 



(4) N'.. = [Jd UdVf{U,V) U' V{ 



— CD 



and for the moment N.. about the mean 



+ 00 



(4*) ■ N,.=ffdUdVf{U,V)(U~-XS(r-Y,y, 



— oo 



wliere denotes the mean value of Ü and the mean value of V. 



We may express the same equations in another way. Let 



M{x) 



denote the mean value of any quantity x, then we have 



\v'.. = M{u'v% 



(5) 



Especially we have 



