620 ON THE PROBLEM OF RANDOM VIBRATIONS, AND [441 as equally probable. The probability that these cosines shall lie within the interval /^ and fa + cfy*a , pz and /A, + dfj^, . . . /AM_I and /u.n_! + cZ/An_j. is !, ........................... (54) which is now to be integrated under the condition that the «th radius s,t shall be less than r. We begin with two stretches Z: and £3. Then, in the same notation as before, we have the integration being AVI thin such limits as make sz < r, where sf^lf + lf- 2Zxi,/*. "Hence, by introduction of the discontinuous function (51), P/ 7 7 \ * J 71 7 [M / / \ ^__ i sv it j CLOG 7Tj_i Jo But by (52) ID ' , ,, -T. , T ,, 2 f°° 7 sinra racosra sm^tf sin4« /r~N and thus -r2(r; ^u 12) = dx------------------------.---- ,----.......(55) A simpler form is available for dP?Jdr, since -7- (sin rx rx COST#) = r#2 sin rx. dr ' ' Thus _^ = _ - sin rx sin l^x sin lzx, ...............(56) .in which we replace the product of sines by means of 4 sin rx sin l^x sin lzx sin (r -f la - Za) x +' sin (r 12 + Z,) « sin (r -f Z2 + ^) a; sin (r ~ L ^) x. . If r, Z2, h 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 /"*" sin 24 7 , =" aM==A-7r; - Jo u ' we obtain three positive and one negative term. Finally in agreement with (47). The expression is applicable only when the triangle is possible. In the contrary case we find dP/dr equal to zero when r is less than the difference and greater tlian the sum,of ^ and 12.e in one plane. Instead of the angles 0, we have now to deal with their cosines, of which all values are to be regarded