32S Lord Rayleigh : Problem of Random Vibrations, 



isolated points by a continuous representative curve. The 

 difference between the abscissae of consecutive isolated points 

 is 21; so that if dx be a large multiple of I, we may take 



VitD^ 2 "'^ 1 ' • • • • (is) 



as the approximate expression of the probability that the 

 resultant amplitude lies between x and x+ dx. 



Two Dimensions. 



If there is but one stretch of length /, the only possible 

 value of r is of course I. 



When there are two stretches of lengths l x and / 2 , r may 

 vary from / 2 — /] to / 2 + /i, and then if 6 be the angle between 

 them 



r 2 = / 1 2 + / 2 * 2 -2/ 1 / 2 cos6>, .... (17) 

 and sin 6 d0 = rdr/l l l 2 (18) 



Since all angles between and tt are deemed equally 

 probable, the chance of an angle between 6 and 0-+-d0is 

 dOJTr. Accordingly the chance that the resultant r lies 

 between r and r-\- dr is 



Tripsins' K "' 



or if with Prof. Pearson * we refer the probability to unit of 

 area in the plane of representation, 



^^-YnrH^And 



7 rV{2^ 1 2 + y)-/- 4 -(Z, 2 -4 2 ) 2 }' ' 



(20) 



4> 2 {r 2 )dA denoting the chance of the representative point 

 lying in a small area dA at distance r from the origin. 

 If the stretches Z x and I 2 are equal, (20) reduces to 



fr^ gwik-f} ' • • • • {n) 



Prof. Pearson's expression, applicable when r<2l, When 

 r>«, <fc(r*)=0. 



When there are three equal stretches (n = 3), (j>s(r 2 ) is 



* Drapers' Company Research Memoirs, Biometric Series III., London, 

 3906. 



