430 Mr. Bowditch on the motion of a pendulum 
the four quarters of the parallelogram are exactly similar and equal, 
The pendulum will return to its point of projection E, in a time equal 
to four vibrations of the pendulum 7, after which it will recommence 
its former course. The same may be observed in other values of ¢. 
When c=135° the equation (I) becomes saa! He 4 
fj 2 = of the fourth ‘order. ‘The body will then describe a curve 
of the form abcdefga (Pl. III. Fig. 12) according to the order of the let- 
ters. ‘This curve is easily described by means of the equations #=6 
cosine (2@¢t+135°) and y=é' cosine a’t, and is exactly like that in Fig. 
10, corresponding to c=45°, except in being placed in an opposite po- 
sition. 
When c=180°, the equation (I) becomes — = =1— 4 or (b—w). 
ayy, which is the equation of a parabola ASC (PI. IIL. Fig. 13) 
whose axis is NS, vertex S, absciss 6—a, ordinate y ; being exactly 
similar and equal to that in Fig. 9, (corresponding to c=0) _— 
in an opposite situation. 
By taking ¢ greater than 180° and less than 360°, we obtain nonew 
curves; the form being precisely the same, whether we use the: value 
¢ or its supplement to 360°; but there is this essential difference, that 
the curves are described ina contrary order. Because when t=0, this 
change in the value of ¢ produces a change of sign in the value of dx 
in the equation (C), the value of dy remaining in both cases =% Thus 
when c=225°, the curve is as in Fig. 12, described according t° the 
order of the letters gfedebag. When c=270°, the curve is aS in Fig- 
LL, described in the order EgGfWadGcE, When c=315%, the euve 
is as in Fig. 10, described in the order gfedebag. 
Hence it appears, that when e=o the curve has but one vertex N (Fig. ; 
9); as chara amaee this opens into two vertices, which gradually separate 
