154 
Proceedings of the Royal Society 
given out by a similar bar of tbe same length held firmly at each 
end. The equation of the curve in this case is 
i h = Eh • 0 j - 0p 2 • 0 j ■ 
Cases IY. and Y. 
are shown to be included in the two preceding ; the touched ex- 
tremity corresponding to the middles of the lengths in those cases 
when the node of the curve happens to be there. 
Case YI. 
In this case, when the ends are held, but without angular tension, 
the equation of condition is 
07-0T- 0 - 
but the difference between these two quaternary functions is the 
X 
sine of — , and so the equation takes tbe familiar form 
V 
. X n 
sin — = 0 
c 
x 
and so — has an infinity of values forming the arithmetical pro- 
L 
gression 7 r, 27 t, 3tt, 47 r, &c. ; and the equation to the curve becomes 
□ 0C- i 1 '%Aj • 
7 “ 0 7 = sin 
X 
X 
i ’ 
so that in vibrating, the bar takes the form of the curve of sines, 
and the nodes divide the whole length equally. In this case the 
rapidities of the different vibrations are as the squares of the series 
of natural numbers. 
The determination of the numerical values of the coefficients in 
the other cases must be obtained by the solution of the transcen- 
dental equations 
[T]p 2 - Qp 2 = - 1, and Qp 2 - = + 1 , 
and this solution is obtained in the following manner. 
By putting Ox to represent the complex function [oJa? 2 — [Y| x?, 
and taking its successive derivatives, we obtain, for the fourth de- 
rivative, the result 
4 xOx = --4 Ox , 
