and Random F tig 1 its in one, two, or three Dimensions. 329 



expressible by elliptic functions * with a discontinuity in 

 form as r passes through I. 



For values of n from 4 to 7 inclusive, Pearson's work is 

 founded upon the general functional relation f 



4> n+1 (r 2 )=l^<P,y + P-2rl C os8)dt). . . (22) 



Putting r = 0, he deduces the special conclusion that 



*. + i(0) = *„(P), (23) 



as is indeed evident a priori. 



From (22) the successive forms are determined graphically. 

 For values of n higher than 7 an analytical expression 

 proceeding by powers of 1/n is available, and will be further 

 referred to later. 



A remarkable advance in the theory of random vibrations 

 and of flights in two dimensions, when the number (n) is 

 finite, is due to J. C. Kluyver J, who has discovered an 

 expression for the probability of various resultants in the 

 form of a definite integral involving Bessel's functions. 

 His exposition is rather concise, and I think I shall be doing 

 a service in reproducing it with some developments and slight 

 changes of notation. It depends upon the use of a discon- 

 tinuous integral evaluated by Weber, viz. 



1 : 



Ji( bx)J Q (ax)dx = u (say). 

 To examine this we substitute from 



7r . J^&a?) = 2 1 cos sin (bx cos 6)dS §, 

 and take first the integration with respect to x. We have || 



Jdx sin (has cos 6) J Q (ax) = 0, if a 2 > b 2 cos 2 0, 



o 



or =(b 2 cos 2 0-a 2 )-% if b 2 coJ6>d\ 



Thus, if a 2 >b 2 , u = 0. If b 2 >a 2 , 



ddcosd 2 . ^ ft sin 





cos 2 e-a 2 ) ~ 7rb Sm V(b 2 -a 2 ) 



* Pearson (/. e.) attributes this evaluation to G. T. Bennett. 



t Compare ' Theory of Sound/ § 42 a. 



X Amsterdam Proceedings, vol. viii. p. 341 (190o). 



§ Gray and Matthews, ' Bessel's Functions,' p. 18, equation (46). 



|| G. and M. p. 73. 



