for a System of Correlated Variables. 7 



^»° • °° • 



i ^in x \ *»in t 



13. Now, I - dxh evidently positive, and 1 - — - dx is 



«^0 J 7T 



evidently negative. Therefore there exists an angle z 

 between zero and 7r such that 1 dx=0, and similarly 



Jm' n T C sin x 7 t sin a? 7 



— <Z#=0. But i — - dx = — - -<te 

 X J & * y _ 57 



— » -* _0 ° 



is a determinate positive quantity, which shall be denoted 



b 7 Z - 



f °° ' f sin* 



Again, ?^ da? is negative, and S — da? is posi- 



tive. Therefore there exists an angle Z\ between ir and 27r, 

 such that ^^dx = 0, and therefore also «£f ^=0 

 Similarly there is an angle r 2 between 27T and 37T such that 

 I sin lV dx = Q. And the range of integration from z to go , 

 or from — z to — x> , may thus be divided into parts z x — z 

 z 2 —z u &c., such that — - d.v = Q for all integral values 



of q greater than unity. 



14. It can now be shown that 



*q-l 



$(*i---0 being by virtue of the equation (s n — r n )A = x 

 a function of x. 



For we may suppose SB to increase from -co to +<x> 

 by successive increments each not greater than 2ir. Then 



for any such increment of x, the increment of s n is = ^- 



that is, it is infinitesimal. Also since (p(s l . . . s n ) is finite and 



continuous for all possible values of s^ ... s n , ^ 



its >i 

 cannot be infinite. It follows that corresponding to the 



