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



operating factor by (l + <J) 3 , integrating the transformed equation, and then making & vanish. 

 It is 



F(r) = Ar- 1 + Br + Cr log r + Dr 3 , (130) 



which gives 



R = — \ = (Ar~ 2 + B + C logr + Dr 2 ) cos 9, 

 rd9 



9 = - -^ = (Ar~ 2 -B-C-Closr- 3Dr*) sin 9. 

 dr 



The first of the equations of condition (117) requires that 



C = 0, D = 0, B = - V, 



which also satisfies the second. We have thus only one arbitrary constant left whereby to 

 satisfy the two equations of condition (116), and the same value of A will not satisfy these 

 two equations. 



46. It appears then that the supposition of steady motion is inadmissible. It will be 

 remembered that, in the case of the sphere, the solution of the problem was only possible 

 because it so happened that the values of two arbitrary constants determined by satisfying the 

 first of the equations of condition (117) satisfied also the second, which indicates that the 

 solution was to a certain extent tentative. We have evidently a right to conceive a sphere or 

 infinite cylinder to exist at rest in an infinite mass of fluid also at rest, to suppose the sphere 

 or cylinder to be then moved with a uniform velocity V, and to propose for determination the 

 motion of the fluid at the end of the time t. But we have no right to assume that the motion 

 approaches a permanent state as t increases indefinitely. We may follow either of two courses. 

 We may proceed to solve the general problem in which the sphere or cylinder is supposed to 

 move from rest, and then examine what results we obtain by supposing t to increase indefi- 

 nitely, or else we may assume for trial that the motion is steady, and proceed to inquire 

 whether we can satisfy all the conditions of the problem on this supposition. The former 

 course would have the disadvantage of requiring a complicated analysis for the sake of ob- 

 taining a comparatively simple result, and it is even possible that the solution of the problem 

 might baffle us altogether ; but if we adopt the latter course, we must not forget that the 

 equations with which we work are only provisional. 



It might be objected that the impossibility of satisfying the conditions of the problem on 

 the hypothesis of steady motion arose from our assumption that sin 9 was a factor of ■%, the 

 other factor being independent of 9. This however is not the case. For, for given values of 

 r and t, ^ is a finite function of 9 from 9 = to 9 = w. We have a right to suppose ■% to 

 vanish at any point of the axis of x positive that we please ; and if we suppose ^ to vanish at 

 one such point, it may be shewn as in the note to Art. 15, that ^ will vanish at all points of 

 the axis of x positive or x negative. Hence ^ may be expanded in a convergent series of sines 

 of 9 and its multiples; and since ^ and its derivatives with respect to 9 alter continuously 

 with 9; the expansions of the derivatives will be got by direct differentiation*. This being 



See a paper On the Critical Values of the Sums of Periodic Series. Camb. Phil. Trans. Vol. VIII. p. 533. 



