suspended from two points. 427 
16. From what has been said in the last article,-we can easily 
: a al 
trace the curve described when o is small, and for illustration we 
shall take the case when the body falls from rest from the point K 
(Fig. 2). The arch 4 becomes (a—a’) t, because by the equations 
(F) ¢ and ¢’ are then =0; and in using the first method (Fig. 7), we 
have SL=0' cosine (a—a ) t, and GR=0' sine c—a’.t. At the com- 
mencement of the motion when ¢=o, the direction of the body will be 
in the right line KI (Fig. 2), which is tangent to the curve in K, as 
was observed in § 8, and this line nearly coincides with the diagonal 
KA, the semidiameter GR €§ie. 7) is 0, and it rapidly increases in the 
successive revolutions. After 37 vibrations of the pendulum of the 
length 7, the arch (a—a) t becomes 90°, then GL=4 and GR=V’, and 
the ellipsis becomes as in Fig. 8, the motion being in the direction 
ESWN. The semidiameter GR (Fig. 7) will then decrease, and 
after m revolutions the arch c—a’. ¢ will become 180°, GR will be o, 
SL=—J’, and the point L will fallin D, the motion being then in the 
straight line DI (Fig. 4), soreeponeng pearly. with, the pines et cna Ds 
nal DC. The body when D will be at r ‘the equa 
tions (C), ¢ and ¢ being 0; at=a't+180°, and m 7 elie sappésed a 
whole number. From this point the body will begin to fall in direc- 
tion of the line DI, the expression of GR (Fig. 7) will become nega- 
tive, and independent of its sign will rapidly increase, and the motion 
will become elliptical. After 3m vibrations the arch c—a’ . ¢ will be- 
come 270°, GL=é and SL= Diy and the ellipsis will become as in 
Fig. 5, the motion being in the direction SENW, contrary to what it 
~ was when the number of vibrations was $e which is conformable. to 
the equations (C), which show that at the sop W, (where: the 
curve touches the line CK) £?<0, and that = eo its sign in 
