426 Mr. Bowditch on the motion of a pendulum 
The jirst of these methods is best to be used when 6 exceeds 6, 
the second when b’ exceeds 8. It is evident that in both methods the 
sum of the squares of the conjugate semidiameters will be equal to 
66+8'b', and the area of the parallelogram included by the tangents 
drawn through the extremities of the diameters will be 400° sine 
(a—a’.t-+e—c'). 
The greatest value of the semidiameter GL or GL’ corresponds 
to the case where SL=+4' or WL=+4, or a—a’ . ¢+e—e'=0° or 180°, 
For then GL, GL fall on one of the diagonals AK, CD, of the paral- 
lelogram ; the conjugate semidiameter GR becomes 9, and the ellip- 
sis changes into one of these diagonals. The semidiameter GL of 
the first method cannot be less than GS, that of GL’ in the second 
method not less than GW. : 
15. In the time of one vibration of a simple pendulum of the 
length r the arch a¢ will increase 180°, by § 9, and the arch A=a—a'# 
to—e will increase in the same time 180°. —— ase ee 5M by § 14: the 
archh will therefore i increase 360° in 2m a ics i or m revolutions } 
of the same pendulum, (or 2m—2 vibrations of a simple pendulum of 
the length t+1') and in that time the conjugate diameters of the above- 
mentioned ellipsis will go through all their changes of magnitude and, 
if m be a whole number, the compound pendulum will be at the same 
place as at the beginning of the time, consequently the whole cycle of mo- 
tions will be complete, and the body will again begin to describe the same 
curves as in the former period. The same thing takes place nearly 
when mis large, but not a whole number, in taking m’ for the nearest 
whole number to m: for, in this case the curves described after sad 
revolutions of the simple pendulum r, will be nearly like those in the 
preceding period, as when m is a whole number. 
