of a Number of Unit Vibrations. 511 



tind we begin by supposing a. less than ir, so that the extreme 

 values of r are 2cos|a and 2. We proceed to consider the 

 relations by which the limiting values of r and 6 are 

 -connected. 



For a given (positive) 6 less than \a. the upper limit of r 

 is 2 and the lower limit is 2cos(-Ja — 6). When 6>^a, 

 there are no corresponding values of r. In fig. 2, where a 

 is taken to be \ir t the shaded area gives the possible values 

 of r corresponding to any #, or conversely the values of 6 

 corresponding to a prescribed r. 



In order to find the chances of a given 0, we integrate 

 with respect to r in (29). We find 



AdOC* dr Add 



^Lo.a.-*,^^^"-^' • (31) 



as the chance that #, if positive, lies between 6 and 0-\-dd. 

 If we integrate (31) again with respect to 6 between and 

 J«, we get £-, the correct value, as there is an equal chance 

 of 6 being negative. 



Again, in order to find the chance of a prescribed r, when 

 6 is free to vary, Ave have to integrate (29) first with respect 

 to 6. Referring to fig. 2, we see that when r<2 cos (Ja), 

 there are no corresponding values of 6, and that when r 

 lies between 2cos(^«) and 2, the limits for d are and 

 \a— cos -1 (^r). In the first case there is no possibility of r 

 lying between r and r + dr; in the second case the proba- 

 bility is 



\.dr 

 g ^(4_ r 2) H"-cos '(ir)}, . . . (32) 



which must be doubled when we admit, as we must, negative 

 values of 6. If we integrate (32) as it stands, again with 

 respect to r, we find the correct value, since 



'2cos|a«V(4— r-) 



{ l a _ C0S -l(l ? .)J_! 



2- 



We may regard (31) and (32) as the solution of the 

 problem in the case where a<7r. 



When a>7r, 6 may lie outside the limits ±Ja applicable 

 to #x and # 2 , and the question becomes more complicated. 

 It appears that we must distinguish two cases under this 

 head, (i.) where 7r<a<37i72, and (ii.) where 37r/2<a<27r. 



First for 7r<a<37r/2, fig. 4, where « is supposed to be 

 -5tt/4. 



