of a Number of Unit Vibrations. 513 



From = to # = !(« — 7r), r ranges from to 2. From 

 = i(a — 7r) to = ±ot, r ranges from 2 cos (■£« — 0) to 2. At 

 6 = \a. the lower and upper limits coincide. From = \ol 

 to #=37r/2 — Ja, there are no corresponding values of r. 

 At the latter limit a zero value of r enters, and from 

 = 3tt/2 — -Ja to = ir, r ranges f rom to 2 cos (2ir - ±* - ) . 



The whole range from = to = ir thus divides itself 

 into four parts. In the first part from 6 — to # = i(a — 7r), 

 we get as the chance of from (29) 



4_d0f 3 



dr 2irJ6 



Jo •(4-0 



2 1 ./CI _„-'! „2 



(33) 



In the second part from 0—\(a — if) to = Ja, tl 

 chance is 



le 



±ddC 2 



^ J 2 cos (%a- 6) 



4 ^(i*-c?). . (34) 



i/(4-r 2 ) 



For the third part, from = la to 0= 3-77-/2 — -Ja, there is 

 no possibility. 



For the fourth part, from 6 = 3ir!2 — \a. to = it, the 

 chance for 6 is 



If we integrate (33), (34), and (35) over the (positive) 

 ranges to which they apply and add the results, we get the 

 correct value, viz. -|. This part of the question might he 

 treated more simply without introducing r at all. 



We have next to consider what in this case, viz. ir<a< %irl2 y 

 are the probabilities of various r's when 6 is allowed to vary. 

 When r is less than its value at = ir, viz. 2 cos (ir— fa), 

 the corresponding range for is made up of two parts, the 

 first from 6 = to = ^oc— cos -1 (\r) , and the second from 

 0=27r — Ja — cos -1 (ir) to = 7r, so that the whole range 

 of is 



^a — C0S _1 (^r)4-7T— {27T — \a— cos -1 (lr)}=a— 7T. 



Thus from r — to r = 2 cos (it — \ol) the chance of r lying 

 between v and r + dr is 



4rfr(a — 7r) _. 



«V(4-0 ( } 



When r lies between 2 cos (tt— Ja) and 2, the second part 

 disappears and we have only the one range of 0, equal to 

 Phil. Mag. S. 6. Vol. 37. No. 221. May 1919. 2 N 



