340 Lord Rayleigh : Problem of Random Vibrations, 



regarded as equally probable. The probability that these 

 cosines shall lie within the interval /^ and /j,i-\-d/n^ /z 2 and 



— l d [ x l d f x 2 ....dfx n _ l , .... (54) 



which is now to be integrated under- the condition that the 

 ?ith radius s n shall be less than r. 



We begin with two stretches l x and L. Then, in the same 

 notation as before, we have 



P s <>; li,h)=b$dii, 



the integration being within such limits as make s 2 >r, 

 where 



, 2 2 = / 1 2-f/ 2 2_2/ 1 / 2 ^. 



Hence, by introduction of the discontinuous function (51), 

 -r,/ j j, 1 I +1 7 T w 7 sin s. 2 x sinrx — rxcosrx 



But by (52) 



x sin Lx 



, i +1 , sm s^ sin ka 

 J_i n S 2 X l^x 



l 2 tt 

 and thus 



r, , 7 7 2 T" . sinra? — r.r co^-a? sin Zi^ sin l 2 x , rrx 



A simpler form is available for d~P 2 /dr, since 

 d 



. (sin rx—vx cos ra?) =rx 2 sin r#. 



Thus 



dF< 



— rrl — sin ra? sin /la? sin / 2 #, . . (56) 

 7rZi/ 2 J * 



in which we replace the product of sines by means of 



4 sin rx sin l x x sin Z 2 *i? = sin (r + Z 2 — ^i) <# 



+ sin (r — Z 2 + /i)# — sin (r + 4 + /i) a? — sin (r — l 2 — l^)x. 



If r, 7 2 , l\ are sides of a real triangle, any two of them 

 together are in general greater than the third, and thus 

 when the integration is effected by the formula 



4 sin u 



Jo u 



sm u , , 



dU = jjTT, 



