[42] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 

 which becomes by means of the differential equation (85) which F 3 satisfies 



F-^'cn^—^-^WX^in,. ..... (98) 



[ m*aF 3 (a)] 



Let 



''•' a^ '*-^"^ (99) 



m ar 3 {a) 



where k and k' are real, then, as before, kM' ——■ will be the part of F which alters the time 



or 



dp 

 of oscillation, and k'M'n— the part which produces a diminution in the arc of oscillation. 



at 



When (i! vanishes, m becomes infinite, and we get from (88) and (99), remembering that 

 D'= 0; k = 1, k'= 0, a result which follows directly and very simply from the ordinary equa- 

 tions of hydrodynamics*. 



32. Every thing is now reduced to the numerical calculation of the quantities k, 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 m a 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 2tn be the modulus of ma; then 



- V^T a / n a / ir 



ma = 2tne 4 , m -r V - - ~ V ~ > • • • (100) 



2 v fi 2 /xt 



t being as before the time of oscillation from rest to rest. Substituting in (99) the above 

 expression for ma, we get 



j ' ,/ \/ - 1 aF 3 '(a) 



Putting for shortness 



log e 4 + *--*r'(l) = - A (102) 



we get from (87) and (93) 



^F,(a)-(logm + A + ^^)( 1+ ^yTl-JnL-_|l_ v /3T + ... ) 



• See Camb. Phil. Trans. Vol. VIII. p. 116. 



