418 Mr. Bowditch on the motion of a pendulum 
If the body fall from a point atrest, we shall have v=o, v'=o, 
whence, by the equations (D) d=-+e, b= +e’, c=o,c'=o, consequently 
a =6 cosine at 
y SEQUISS (F) 
y= 0 cosine at 
These last equations may a/vays be used when a, a’, are incom- 
mensurable, which includes by far the greatest number of cases. 
For if the arch at be decreased by any whole number f of circles re- 
ae by p. 360° the arch a¢ would be decreased cay pre “7 
a? 
a—a 
become 4 cosine Btto—p 360". ty and if a ka were incommen- 
surable, we might take p ach that p. 360% would inneglecting mul- 
tiples of 360°) be equal to ¢, or differ from it by a quantity less than 
any assignable, which would reduce the expression of x to 6 cosine af, 
taking for epoch the time corresponding to this value of p. But when 
a@and a’ are commensurable, this reduction can take place vege for 
particular values of ¢, which it will not be necessary to note, since 
they may be easily discovered. 
6. The equation 0=1 + /rr—ze—V/ zz4+yy Of $4, esas in 
series, neglecting the fourth powers of x and y gives z=r He 
x” 
Poe which by substituting the values of x, y, given in the equi- 
c B t iaieth 
tions a béconiés zertl ae, cosine at+c) a?) : 
a we have bedava approximate values of a, Ys 25 seit? 
ing to the time ¢, which will enable us to trace nearly the course of 
the pendulous. body; and if greater accuracy were required, We 
might substitute the values of Z, y, in the third equation (A). and. 
thus obtain a more correct value of A which substituted in the ‘other 
two equations would give exacter values of a, y, and in this way, by 
