622	ON THE  PROBLEM OF RANDOM VIBRATIONS, AND	[44!
d2 /IdPA      2   (™dso .         .  .,
and	- T~a (- T- ) = ~74      ~ sm rx sm lx
dr* \r dr /     nrL* J0   x
— sin?' + 4j# + sin?^-4/) x
4 sin (r + 2£) # — 4 sin (r — 2£) a? + 6 sin rx}
the alternatives depending upon the signs of r - 4£ and r — 2£.
When	r<2£,    - 16/^2 (- -^} = 6,
dr2 \r dr J
dr J	'
and when r > 4>l, the value is zero.    In no case can the value be infinite, from which we may infer that
c?r /	?• dr
must be continuous throughout.
From these data we can determine the form of dPJdr, working backwards from the large value of r, where all derivatives vanish.
\	/	dr\rdrj	^         '
(21 >r)    - W 4- (- ~r~} = 6 (r - 21) + 4,1 = Qr - SI,
^	dr \r dr)	'
giving continuity at r = 4<l and r = 21.    Again
(4,1 > r > 21)    -16/4 i —^ = - (r3 - 16?) + 81 (r - 4<l)
(£(j  ^   I J          "~~  JL 0 v                 J         •"•"•  O  ( /       ""* ~l}v   ) ~~  O v ( 7* ""~°"  £iu) *™~" Toy
——  QrttS  ,	rN')'*/
Finally      4r = 4^2^(r; 0--^—^    (r<2Z)
and vanishes, of course, when r > 4>l.
From (61), (62) we may verify Pearson's relation* fa (o ; 1) = </>3 (I ; 2).
[* This implies that Pearson's relation (p; 610) holds for three dimensions.   We have in fact, for flights in three dimensions,
6 ; I) sin Ode,
whence 0n+1 (o ; 1) = ^n (Z ; ?.).   W. P. S.]