and Random Flights in one. two. or three Dimensions. 343 

 giving continuity at r — ll and r = 2f. Again 



/■ dr 



= -{r-±iy, 



{U>r>U) -16/ 4 ^^p= - (r*-l6F) + $l(r-4:l) 



(2l>r) -16/ 4 - d ^ = 3fr ? -4/ 2 )-S/(r- 2/)-4/ 2 



= 3;- 2 -8r/. 



Final 1 



= 47r/-^(r; /)=- (r<2/) 



rf4 1 rV >> (62) 



16/' 



J 



and vanishes, o£ course, when r>4/. 



From (61), (64) we may verify Pearson's relation, 



From these examples the procedure will be understood. 

 When n is even, we differentiate (59) (ii — 2) times, thus 



obtaining 



in which sin" /a,' is replaced by the series containing cos?i/#, 

 cos (n — 2)/a', .... and ending with a constant term. When 

 this is multiplied by sin ra; we get sines of (i' + nl)x, 

 {r + (n — 2)l}x, .... sin ra?, and the integration can be effected. 

 Over the various ranges of 21 the values are constant, but 

 they change discontinuous] y when r is an even multiple of /. 

 The actual forms for dP„jdr can then be found, as already 

 exemplified, by working backwards from r>nl, where all 

 derivatives vanish, and so determining the constants of 

 integration as to maintain continuity throughout. These 

 forms are in all cases algebraic. 



When n is odd, we differentiate (?i — 3) times, thus obtaining 

 a form similar to (60) where n = 3. A similar procedure 

 then shows that the result assumes constant values over 

 finite ranges with discontinuities when r is an odd multiple 

 of /. On integration the forms for dP n /dr are again 

 algebraic. 



I have carried out the detailed calculation for n = 6. It 

 will suffice to record the principal results. For the values of 



2 6 / 6 



dr 4 \r dr J 



