of Edinburgh, Session 1866 - 67 . 153 
the zero being placed at the fixed end, we must have A = 0 , B = 0 , 
while the coefficients C and D must satisfy the two conditions 
C0y + D[i]j = O; 00^ + DQf = 0; 
which give 
rn X ^ x rn X rn X 
DD T • 0I = 0J • 0 T ; 
from which equation the ratio -j has to be determined. 
Denoting this ratio by p 1? we observe that p v has an infinite num- 
ber of values corresponding to the infinite number of simple vibra- 
tions of which the system is susceptible, and that the times of 
these vibrations are as the squares of the different values of l thence 
resulting. This general equation of condition may also be put 
into the form 
0Pi 2 - Bpp = ~ 1 ; 
and the equation of the curve becomes 
yi - 0pi • El | - Eft • E i ■ 
Case II. 
When both ends of the bar are free. 
In this case the equation of condition becomes 
0P* • 0Pj = • Bp2 > 
or 0P/ - Hft* = + 1 . 
while that of the curve is 
& =■ Sp, • □ f - Eft-Ef- 
Case III. 
When both ends are held fast. 
The equation of condition in this case is found to be identic 
with that of the preceding case, and hence we have this very re- 
markable law, that the sounds emitted by an elastic bar when 
suspended so as to vibrate freely, are identic in pitch with those 
VOL. VI. 
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