96 



C. V. L. Charlier 



Putting 



T] = cotg '\ , 



we get 



N — N ' 



^'20 -^'02 



Comparing this formula with the formula (74) above, we conclude that <\i = cp,, 

 = the angle between the line of symmetry and the axis of Ü (= X). 



The equation (97) gives the value of the quotient between and JV^^. The 

 values of iV^ and are then given by the equations 



From (92*) we get 



1 +P 



'lY,, NN 



n. — = + 1/ , 



(98) ' ^ " r vj N^N^ ' 



ni^ — ^2 ^ (^1 — '^a)- 



The relations (91) and (91*) give us now the values of m^, m^, w^, n^. 



It remains to determine the dispersions ot^ and a^. This determination can 

 take place with the help of the relations (92) and (93) or from one of the equations 

 (94) combined with (92). In the former case we have to deal with two linear 

 equations in a^^ and a^^, in the letter case we have a quadratic equation in a^^ — a^^ 

 combined with a linear equation in a^^ and a^^. Both methods must lead to 

 (approximately) the same values of the dispersions. 



As to the choice between the two roots of the equation for t and between the 

 signs of (98), we have to take recourse to the equation (96*), which is sufficient for 

 the purpose, taking into consideration that the choice of the indices for the two 

 streams is arbitrary. We may for instance give index one to the northern stream, 

 in which case n^ > w, and the positive sign in (98) must be chosen. From (96*) 

 we then are able to conclude whether N^ > N^ (h. e. > \) or whether N^ < N^ 

 (h. e. f < 1). 



The dissection of the correlation surface into two circular streams is hereby 

 completed. 



31. The solution given above of the problem to dissect a given correlation 

 surface into two sphserical streams can be applied even for dissecting a given fre- 

 quency curve into two normal curves. As well known, the solution of this problem 

 has been given by Pearson, but necessitates the computation of the moments, of 

 the fifth order and leads, moreover, to a resultant equation of the ninth degree, 

 whereas above only equations of the second degree, at most, occur and only moments 

 of the third degree, and lower ones, are wanted. On the other side we must here 

 know the frequency distribution of two characters of the statistical object. 



