450 Slichter—Harmonic Curves of Three Frequencies. 
approximately in line with the two steel points which support 
the pendulum P. If the pendulum bob P be adjusted so that 
the pendulum has the third desired frequency, then when the 
pendulum P is vibrating, the image of the lamp L will describe 
upon the sensitive plate of the camera harmonic ^notion of the 
desired frequency. To secure a photograph containing a curve 
possessing all three of the frequencies, the bob of the Black¬ 
burn pendulum is held at one corner of the table by the elec¬ 
tro-magnet E' and, at the same time, the camera pendulum is 
held at the end of its swing by the electro-magnet E. The elec¬ 
tro-magnets are on the same circuit and are controlled by the 
key K'. The key K is then pressed to illuminate the electric 
lamp L, and immediately afterwards the pendulums are released 
by the key IT. At the close of the complete period of the com¬ 
pound harmonic curve, the key K is released, extinguishing the 
light L. 
There are given in figures 1-20 copies of stereoscopic photo¬ 
graphs taken by means of the apparatus above described. The 
camera could be placed on either of two adjacent sides of the 
table and two views made of the same curve, if so desired. Fig¬ 
ures 1 and 2, also 17 and 18, present views of the same curves 
taken from adjacent sides of the table. Figure 14 shows the 
same curve as figure 13, but is taken from a point opposite the 
corner of the table. 
Mr. Elting H. Comstock, at that time a senior in the Uni¬ 
versity of Wisconsin, succeeded in working out, by an original 
method, the number of intersections or double points in the 
plane harmonic curves, and, by the same method, succeeded in 
obtaining the number of double points in harmonic curves of 
three frequencies. The formulas obtained by him are as follows: 
Let n: r: s be the ratio of the periods of the component har¬ 
monic motions. Let a be the highest common factor of r and s, 
P the highest common factor of s and n , and y the highest 
common factor of n and*r. The number of double points is then 
given by 
(n __l )(a _ 1) I (r — 1 )((5 — 1) , (s-l)( r -l) 
2 * 2 2 
_ («-!)(/?-!) __ (yg-l)(r-l) __ (y-l)(a-l) 
2 2 2 
( 1 > 
