in the Theory of Probability. 749 



To do this we differentiate equation (2), this gives 



^(i-iy( <+ < 1} =s rt -T rW . 



' r=— oo 



Therefore 



Differentiating again arid putting £ = 1, we get 



r= -co 



Hence the probable value of r 2 is v. 



A somewhat similar result is obtained in B-ayleigh's 

 ' Sound/ vol. i. p. 36, where it is shown that i£ n unit vectors 

 whose signs can be either positive or negative, are combined 

 so as to give a resultant of magnitude r, then the probable 

 value of r 2 is n. 



3. To find the most probable value of r we take the 

 recurrence formula 



Since I„(v) is always positive, we have 

 I n _ 1 >I n+ i, n>0. 



Also 



Consequently, 



VW=*.[i-i+Ui3; 



I.'OO < I»-i(v) and I„'0)>I„+iO). 



In -i(v) !.'(>) WO 



1»0) ^ I„0) I»(») ' 



then I»-iW >I»W; 



I»W 



These inequalities show that if " ~g 1, 



while if t4^t<1, we have 



U(V) I a+1 (>) , 



i„(>0 i»W • 



and so 



!„(„)> P +1 (i/) ( 



