191 
1906-7.] Dr W. Pecldie on Vibrating Systems. 
into, and its emergence from, the semicircle, we have the 
expressions : 
CA 
2p+l 
a ■ pir+20 
cos 0 — sm [ 
!(/>+!) 
CA 2 „ cos 0 = CB 2i0 cos 0 = sin ~ - , 
p + 1 2 
CIV, cos 6 — sin 
pir - 20 
2{p+i) 
following 
Twice the probability that the emergent path shall pass nearer to the 
centre of the circular area than the entrant path passes is represented by 
2 
(P+1)tt + 20 
2(p+2) 
— sin 
(p - 1 )?r+ 2l 
+ sm 
20 
When n has a given value, 0 lies between 7 r/ 2 n and tt /2 (w-l). 
