suspended from two points. * 431 
tilt they come to the points C, A, Fig. 13. In like manner the points 
D, K of the curve (Fig. 9) form the vertices g, d, as ¢ increases; and 
these points gradually approach till they meet in S, when c=180°. 
While ¢ increases from 180° to 360°, these points go back in the same 
order as they advanced, and when ¢=360°, the curve becomes as in 
Fig. 9, where c=o. 
18. When 2a’ is nearly equal to a, we may put — += 24+-, m be- 
ing a aa number, and since (by )5)— act 
, consequently 7’ must baa Sr, by a very small quantity 
, we shall have nearly 
= IF 
to render m positive. If we suppose the pendulum to be let fall from 
rest from the point K (PI. IIL. Fig. 9) the —— (F) will give y=0' 
cosine a’t and x=. cosine at=. cosine (2a't4.— a) Hence if we put 
C= — and take at successively =0°, 2x 180°, 4x 180°, 6x180°, &c. 
the values of y corresponding will be constantly equal to 4’, and those 
of a will be successively 6, 6. cos. 2C, b. cos. 4C, . C8 6C, &c. and, 
Po abe ee oe oo oe oe Por Meee ngth ; + r', th Pm e value 
after n vibrations of a s imple fp of the 
of x will become 4. cosine 2nC, or be cosine (360°. =), Hess it is 
evident that at the commencement of the motion the body will begin 
to vibrate backwards and forwards in the parabola KND, as in the last 
article, when e=o (Pl. III. Fig. 9), but after a few revolutions the arch 
2nC will increase so much as to make the pendulum return to the line 
KC ata point e, Fig. 10, sensibly different from K, and falling be- 
tween KandC. Thus when n=im (which corresponds to $m vibra- 
tions of the simple pendulum r+7’) the arch 2nC will be 45°, and ue 
curve will be as in Fig. 10, which will be described by the 
according to the order of the letters abedefg, as in the ricci 
cle. When n=im, the arch 2nC will be 90°, and the curve will be as 
in Fig. 11. When =m, the arch 2nC will be 155°, and the curve 
