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Full text of "Scientific Papers - Vi"

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