424 Mr. Bowditch on the motion of a pendulum 
the same resultsa in§9. If cis finite, we must transpose the term 
a 
> cosine ¢, and square both sides, which by reduction will Sie sa 
ee + 
oF 9 eg! a cosine rad a = sine c 
the equation of an ellipsis or circle, when 4 and @ are finite. Wher 
e=90° the equation is reduced to = +55 7 5; =I, corresponding to an dl 
lipsis, which becomes a circle, ate radius is 0, when b=. 
14, When @’ is nearly equal to a, we shall put — =1—-—, aes . 
att , we shall have m 
ing a large number; then, si 
ar s . 
=— nearly, and 7’ will be very small in comparison of 7 This is 
the only case mention in Professor Dean’s paper, and the motion may 
be compared to that of a body ina variable ellipsis. _ For let P (Pl. 
Ill. Fig. 7.) be the place of the pendulum at any moment, when, by 
the equations (B); x=b. cosine (at+e) and y=0' . cosine (atte). This 
last expression, in putting A=z—2’ . t+ce—c’, becomes y= slats 
(at-+c—h) =0'. cosine h. cosine (at+c) +6’. sine ee sine (atte). Ifin 
this we substitute for J’ . cosine (at+e) its value 22, and for Gi. sne 
fi7 ey iV biz fee 
(at+e) its value B, = it will become ate . COS. = 
sine. Therefore, if we take on SK, the line SL=J'. cosine h ad 
draw the line GL to cut the ordinate PM in T, we shall have, 1romh from | 
the similar triangles GMT, GSL, MT.22 — + cosine h, and 
/GL?—GT? 
=" Se Whence y=MP=M T+0' .sine h. 
a TP=i -sineh eae =. Hence, if we take on GW, of 
line GR=0',siic 4, we shall have TP=GR, YOE—S™, which is 
DA a OES i pe ee 
| of bb6— eh aie ponte 
has ce 2 ‘ 
