112 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
P, across to the other side. B is top-heavy so as to lie against a stop, to 
whichever side it is put, H not being required to hold it in place, and only 
coming momentarily in contact with it. The water now runs down the 
waste-pipe K, C gradually empties until at a certain stage the balance tilts 
back owing to the presence of the counter- weight L, and the whole cycle 
of operations begins again. The respective times of filling and emptying 
can be independently adjusted by means of the finely graduated taps at 
A and E. During the filling an electric contact is closed by the dipping 
of a wire into mercury at M, and during the emptying the contact is open. 
The circuit includes a cell ; and an electro-magnet, which working against 
a spring actuates the deflecting tube N (fig. 1), which turning slightly 
about the vertical axle Q, can be made to swing under the nozzle of the 
turbine, and so to carry off the water before the latter reaches the wheel. 
Although a little tedious to describe, the whole arrangement is quite simple, 
and works admirably, the small movement of the deflecting tube taking 
place with great suddenness, so that there is no perceptible splashing of 
the water, or loss of time between the “off” and “on” positions. As the 
balance beam turns on a knife edge, and as it is quite free, H not coming 
in contact with B except during a swing, and even then not until the 
motion is well established, it follows that the timing is very accurate. 
The experimental results will be given after the theory has been 
discussed. 
Theory and Results. 
In fig. 5, if OABC represent one of the starting curves, and DEF 
the stopping curve, then, in order to choose the intervals of time during 
which the water is “off” and “on” so that the motion may be periodic, 
we must move up the starting curve from A to B, and down the stopping 
curve from E to F, so that BE and AF are parallel to the time axis. The 
intervals of “on” and “off*” are then GH and KL respectively, while the 
number of revolutions described during a complete period is represented 
by the sum of the areas GABH and KEFL. The number that ought 
to be described is the terminal velocity multiplied by the interval during 
which the water is “ on.” Assuming the equations already determined to 
give the general shape of the curves of fig. 3, it is easy to show that there 
is always “ overrunning ” for such a periodic motion, and to determine an 
expression to represent it. 
Let 
OG = q , OH = , OK = t 3 , OL = i f 4 (in fig. 5) * 
then, given t ± and t 2 , we must determine t s and to give a periodic motion. 
