( 348 ) 



CO 



Wn {d£^) = -^ ^ dQ dd fu J, {uQ)f{ic) du, 







and here IVn {(l^) means the probability that the ending point of 

 the broken line falls on a given element d^ of the plane, the polar 

 coordinates of which are q, 6. 



By integrating over a given finite region we ma}' dednce the 

 probability that the rambler reaches that region ^). 



First let the region be a rectangle U, and let the rectangnlar 

 coordinates of its vertices be ± ^;, =b c[, then we find for the corre- 

 sponding probability 



Wn {R) = -^ fiifi^^) du Cdi Cdyj J, (u i/rr?). 



-p -q 



Now we have 



271 



Jo {u \/%^ + 'H^) = I COS {u I COS a) cos {u i} sin a) da^ 







and therefore, effectuating the integrations with respect to § and to r], 



n 

 o 



00 



4 r Csin (pu cos d) sin (qu sind) 



Wn {R) = -\ nfyii) du ''\ . ^ '^ da. 



jt\j J u sin a cos a 







A somewhat simpler expression is found, if changing the variables 

 we pass from u and « to 



V z=z u cos «, 

 IV :=■ u sin a. 

 Then the probability TF„ ( E) is expressed as follows : 



u^)=^/^ 



sin pv sm qw 



Wn{R) = — \ 1 dv dw —^ . -^- .f{]/v' + w'). 



V 10 







Again an evaluation of this double integral is generally not practi- 

 cable, but the problem itself gives the value of the integral, if both 



1) If this region is a circle with radius c, the centre of which lies at a distance 

 b from the starling point 0, we have at once 



00 



T^„+i (c; haa, . . . a„_i) = c (j, (uc) J, {ub) J, (ua) J„ na,) ...J, 0<a„_i) du 







for the probability, that the path ends inside the circle. 



