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



inch or half an inch, we have still the square or fourth root of the above quantity, that is, 

 about 49234 or 222, for the value of that exponential. Hence, in practical cases, the above 

 simplification may be made, which will cause the exponentials to disappear from the expression 

 for K. We thus get 



3b (m?a 2 + 3ma + 3) (m c 6 2 - Smb + 3) 



= 1 T 2m?a 2 b (r» 2 6* - 3mb + 3) - a (roV + Sma + 3) ' ' ' ^ 



If we assume 



3va + 3 + (2i> 2 a 2 + 3va)\/ - 1 = -4'(coset + y/-l sina), 



- 3vb + 3 + (Z^b 2 - 3vb) \/ - 1 = 5'(cos /3 + v^-1 sin/3), 



bS cos /3 — a A' cos a = C cos y, 



b fi' sin j8 - a A' sin a = C sin <y, 



we get from (6l) 



3b v — 1 ^i'5' c . . _ . / . , _ . , 



+ Wa* ' ~C~ { c0S ( a + / 3 -7)+V -1 sin(a + j3- 7 )}, 



whence 



, 3bA'B' . , • 



* = , « < ,t^y sln (a + P - 7) ~ J » 



4 1< a C 



^ 36^'^ 

 A;= 4^^C 7COS(a + /3 - 7); 



(62) 



and, as before, kM' is the imaginary mass which we must conceive to be collected at the centre 



of the sphere, in order to allow for the inertia of the fluid, and — k'M'n — the term in F on 



dt 



which depends the diminution in the arc of oscillation. 



23. If we suppose // = 0, and therefore m = co , we get from (61) 



b 3 + 2 a 3 

 2 (o 3 - a 3 ) 



and, in this case, k is the same as K with sign changed, and k' = 0, which agrees with the 

 result obtained directly from the ordinary equations of hydrodynamics*. If, on the other 

 hand, we make 6 = 00 , we arrive at the results already obtained in Art. 20. In both these 

 cases it becomes rigorously exact to neglect in the expression for K — 1 given by (59) all the 

 terms which are not multiplied by e"^""'. 



If the effect of the envelope be but small, which will generally be the case, it will be 

 convenient to calculate k and k' from the formulae (52), which apply to the case in which 

 6 = » , and then add corrections A k, Ak' due to the envelope. We get from (61) 



/ ,, 3 (m ! a 2 + 3ma + 3Y ,„,, 



A k — v — l A k = - ■ (64) 



2m 2 a 6(w» 2 6 3 -3»w6 + 3) - a(m 2 a i + 3ma + 3)' 



See Camb. Phil. Trans. Vol. VIII. p. 120. 



