69 
of Edinburgh, Session 1866-67. 
and the second, almost exactly 4f, exceeds the ratio of the ex- 
pansions of any two available solid substances. Thus we conclude 
that pendulums constructed of two pieces after the manner shown 
in figure 1, cannot be compensated. 
The equation of condition (3) being of an odd degree, must 
have always one root possible. It is true that this is merely a 
numerical possibility, which may or may not be represented by a 
mechanical arrangement; yet it may be worth while to examine 
into it. Now, when n is positive, the equation (3), must have one 
of its roots negative, and thus the point B, corresponding to that 
root, must be the above point of suspension. 
Let us suppose, then, a tube abba of glass or some 
substance having a very slight expansion, to pro- 
ceed upwards from the axis of motion SS, and from 
the upper end of the tube let a heavy rod, BC, of 
more expansive material depend, and the arrange- 
ment will represent, mechanically, the negative value 
of x obtained from the solution of the equation (3). 
On assuming for n a number of successive values, 
and thence computing the corresponding roots of 
the equation (3), as also the values of a , b , and c, 
necessary to give a pendulum vibrating isocbro- 
nously with a simple pendulum whose length is 
unit, and on representing the results geometrically, 
we obtain the diagram given in figure 3. 
In this figure, the distances measured along the 
horizontal line OS indicate the ratio of the expan- 
sibilities of the two substances, 01 standing for 
that of the suspender; 01 or SL is the length of the corresponding 
simple pendulum ; and the curved lines marked B, B 2 B ;1 ; C L 0 2 C 3 
define the values of b and c, corresponding to the three roots of the 
equation. 
For example, if the ratios of the rates of expansion were 9 . 2, let 
us take the point s at that distance along OS, and draw through 
it a vertical line, crossing the six curves at the points b 1 c 3 c 2 , c l b 2 
b 2 ; then sb { c t shows the proportions for a pendulum constructed in 
the manner shown in figure 2. s& 2 c 2 shows that which is intended 
in the actual execution of clocks, while sb. A c ;j is another possible 
Fig. 2. 
