suspended from two points. 429 
will be at D, and will then be at rest, by the equations (C.) When at 
=540°, the body will be again at N, and when at=720°, it will be at 
K. Hence the pendulum will keep vibrating backwards and for- 
wards in the parabolic are KND. | 
= e=45° the equation (I) becomes ; S17 
/1—%, of the fourth order.. The body will then describe a curve 
b/' 
of the form abcdefga (Pl. II. Fig. 10,) according to the order of the 
letters. This curve is easily traced by means of the equations x=) 
cosine (2a't+45°) and y=0'. cosine a’t, deduced from the equations (E). 
The velocity of the pendulum at the point e in the line CK is, by 
the first of the equations (C) —aé. sine 45°, or “} multiplied by the 
velocity acquired by a simple pendulum of the length r in falling from 
the point corresponding to S towards the lowest point G. The points 
a, e, are found by putting y=d' and y=—0’, corresponding to a’t=o 
and a't=180°; which give in both cases x=. cosine 45°=b/}. The 
points g, d, where the curve touches the line DK are found by putting 
a=b, which gives at=315° or 675°, and eye frie mach 
es AC, by putting a=—b, which gives at=135° or OSA; 
When c=90°, the equation (I) becomes J-—4 J 1 1— of the 
fourth order. In this case the body will describe the curve EeGdWf 
GgE (Pl. Ill. Fig. 11) according to the order of those letters. This 
curve will be described if the pendulum be projected from E in the 
direction EA, or from W in the direction WC, with a velocity equal 
to that which a simple pendulum of the length r would. acquire, in fal- 
ling through the arch whose projection is SG. The curve is easily 
traced by means of the values x= —4é sine 2a't, y=U cosine a't, deduc- 
ed from the equations (E). By taking 2=+ we obtain yrttYy 
which is equal to the es Sg, Sd, Ne, Nf. The parts of the path in 
