ON THE MOTION OF PENDULUMS. 47
where k and k' are real, then, as before, kM' d^jdf1 will be the part of F which alters the time of oscillation, and TcM'n d%/dt the part which produces a diminution in the arc of oscillation.
When fi vanishes, m becomes infinite, and we get from (88) and (99), remembering that Z)'=0; & = 1, &' = 0, a result which follows directly and very simply from the ordinary equations of hydrodynamics *.
32. Every thing is now reduced to the numerical calculation of the quantities Jc, k', of which the analytical expressions are given. The series (87) being always convergent might be employed in all cases, but when the modulus of ma is large, it will be far more convenient to employ a series according to descending powers of a. Let us consider the ascending series first.
Let 2m be the modulus of ma ; then
T being as before the time of oscillation from rest to rest. Substituting in (99) the above expression for ma, we get
Putting for shortness
log€4 + ir-*F(i) = -A ..................... (102),
we get from (87) and (93)
m2 / — m4 m6 __ / ~T + -p V - 1 - 1272* - -JT; 22 3a v - i + •••
/m' <, ,— =- m' m° „ /— - \
~ \ ]* "i v I2 22 2 ~ I2 2" 32 ~ ' ' / '
1 -,,., s, ^m2 ,— = w
ffl-P8 (.aj — i + - v — i - j - ...
/m2,, /— Y m4 „ m" g ,-^ \ "llT1 ~ I2 2 2 ~ I2 2^ 3 s "~ ' ' '/ '
* See Camb. Phil. Trans. Vol. TOI. p. 116. [4ret«, Vol. i. p. 37.]