14 



Sven Wickseil 



and 



(25) cT 0 = 9o (x) . ?0 ( Y) . U (Z) [1+2 E i0ü . b 



!+.;+;; > 3 



Having now in a condensed form worked out the theory of frequency- 

 functions of the J.-type for three variables, -I remark that the mode of demonstration 

 is built upon the nice workings of Greiner, and that in all other respects the 

 line of progress is an extension of the elaborations of Charliek regarding one and 

 two variables. 



6. Before leaving the more general discussion of the frequency-functions of 

 the ^.-type I will mention a few theorems. 



The equations 



U-bi |L_0; § = 0 

 dX dY r dZ 



give the coordinates of the »mode». 



Referred to the system of coordinates where the variables are uncorrected, 

 and supposing the coefficients ß^* to be so small that (2 ß^ R ijk ) 2 umy be neglected, 

 we shall find for the coordinates of the »mode» 



X l = 3 ß 3 0 0 + ßl2U + ßlU2 



(26) Fi^3ß 080 +.ß aj0 + ß 012 



%l — 3ß Q Q 3 + ß a Q j + ß o 2 1 • 



The quantities, irrespective of the system of coordinates, 

 Sx = 3ß 30O Sy = 3ß 030 Sx = 3ß 003 

 are called the shewness respectively of x, y and z. 



In the equation (25) we put X = 0, F=0, Z—0 and find by help of (24) 



m 0, 0) = ' fo (0) s [1 -f 3ß 40ü + 3ß 040 + 33 0 04 + ß 220 + ß 202 +ß 0 8 2 ]. 

 Consequently the sum 

 (32**) E = 3ß 400 + 3ß 04ü + 3ß ü04 + ß 2 2 0 + ß 2 0 2 + ß 022 



gives the relative excess over the normal frequency of the points (x, y, s) near the 

 mean. Taking the frequency curves of the coordinates x, y and z separately we 

 have for their excess 



(32*) £r = 3ß 400 #,, = 3ß 040 #z = 3ß 004 . 



The equations (23), (24), (25), (26) and (32**) are only valid for the system 

 of coordinates where the variâtes are uncorrelated. 



