n = k × (e / l2)
in which k is a constant quantity whose value depends upon the nature of the metal of which the tuning fork is made.
This constant varies very little from steel to malleable cast iron, and it may be taken as equal to 818270.
Thus, then, we have a means of constructing a tuning fork in which two of the three quantities, n, e, l, are given in advance. Experience proves that no errors are committed exceeding one or two per cent.
It is seen from this that there is a means of increasing the mass of the instrument without changing anything in the thickness, the length or, consequently, the number of vibrations, and this is by increasing the breadth.
It is in this way that I have succeeded in having long massive tuning forks made of malleable iron, giving no more than 12 to 15 vibrations per second, and vibrating with perfect regularity. Fig. 2, annexed, shows one of these instruments of about 55 centimeters length, whose breadth, E, is from 5 to 6 centimeters, and which makes about fifteen double vibrations per second only.
This number might be still further reduced, but at the expense of our being led to exaggerate the longitudinal dimensions of the apparatus in such a way as to make it inconvenient. The object may be attained more simply by loading the branches with slides supporting leaden weights, M, of 500 grammes each. By fixing these slides at different points on the branches, the number of vibrations can be made to vary from simple to double, and even triple. Thus, by fixing them at the extremity of the branches the number of the vibrations is reduced to 5 or 6.
There will be seen in the figure the electro-magnet which keeps up the vibration. This is formed of three simple electro-magnets, whose bobbins have a resistance of no more than 10 ohms, and which are united in series. The interrupting plate, P, against which the style, s, rests at each vibration, is capable of a forward movement, or one of recoil, by the aid of a screw, V, and of an eccentric movement which is produced by a small handle, m, and during which its plane remains invariable. This arrangement permits the point of contact of the style and plate to be varied without changing the precision with which the contact takes place, and all the points of the plate to be slowly used in succession before replacing it. The motion is produced by means of a relatively weak pile, whose poles are connected to the terminals, A and A'. Three Callaud elements of triple surface, renewed one after the other every month at the most, are sufficient to keep up the vibrations continuously, day and night, without interruption, and that too even when the instrument is employed in producing a small mechanical power, as we shall see further on.
We have now seen how electro-tuning forks may be constructed of large dimensions, of large mass, and giving a small number of vibrations per second.