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50 ON THE CALCULATION OF CHLADNl'S FIGURES FOR A SQUARE PLATE [3f
and Lagrange's equations are
while the equation for p- is the quadratic
pt(Ltf-M*)+>[P(2M£-La-NA) + AC-&=Q ........ (12)
For the numerical calculations we will suppose, following Bitz, that p, = '221 making G = 11 '9226. Thus
Ztf-Jf' = -13714, AC -E2 = 7-9226, 2MB -LC~NA=-2x 4'3498.
The smaller root of the quadratic as calculated by the usual formula i '9239, in place of the 1 of the first approximation; but the process is no arithmetically advantageous. If we substitute this value in the first term c the quadratic, and determine p- from the resulting simple equation, we ge the confirmed and corrected value pz = '9241. Restoring the omitted factors we have finally as the result of the second approximation
in which p = '225.
The value thus obtained is not so low, and therefore not so good, as tha derived by Bitz from the series of ^-functions. One of the advantage; of the latter is that, being normal functions for the simple bar, they allow % to be expressed as a sum of squares of the generalized coordinates q1} &c As a consequence,^2 appears only in the diagonal terms of the system o equations analogous to (11).
From (11) we find further
so that for the approximate form of tu corresponding to the gravest pitch we may take
w = xij - -0852 xy («2 + f), ....................... (14)
in which the side of the square is supposed equal to 2. may omit the factors already accounted for in (10). Expressions (7), (9) are of the standard form if we take