838 Lord Rayleigh : Problem of Random Vibrations, 



Three Dimensions. 



We may now pass on to the corresponding problem when 

 flights take place in three dimensions, where we shall find, as 

 might have been expected, that the mathematics are simpler. 

 And first for two flights of length l^ and l 2 . If At be the 

 cosine of the angle between l x and l 2 and r the resultant, 



r 2 = / 1 2 + L 2 -2/ 1 / 2A t, 



S ivin g rdr^-hkdp (46) 



The chance of r lying between r and r + dr is the same as 

 the chance of //, lying between fi and \x + d^, that is — idfi, 

 since all directions in space are to be treated as equally 

 probable. Accordingly the chance of a resultant between 

 r and r + dr is 



rdr 



2lJ 2 



(47) 



The corresponding volume is kirr^dr, so that in the former 

 notation 



«"9-»s*' < 48 > 



l x and 1 2 being supposed equal. It will be seen that this is 

 simpler than (21). It applies, of course, only when r< 21. 

 When r>2l, 2 = O. 



In like manner when l x and l 2 differ, the chance of a 

 resultant less than r is zero, when r falls short of the 

 difference between l 2 and l lf say l 2 — li. Between l 2 — l x and 

 l 2 +l\ the chance is 



rdr _ ^~(l 2 -ky 



i 



When r has its greatest value (1% + ly), (49) becomes 



(k + hY-ik-hV 



U,l, 



= 3 (50) 



The "chance" is then a certainty, as also when r>l 1 + l 2 . 



In proceeding to the general value of n, we may con- 

 veniently follow the analogy of the two-dimensional investi- 

 gation of Kluyver, for which purpose we require a function 

 that shall be unity when s<r, and zero when s>r. Such a 

 function is 





, sin sx sin rx — rx cos rx ,-:^ s 



dx— ; . . (51) 



sx x v 7 



