64     ON  THE  EFFECT  OF  THE  INTERNAL  FRICTION  OF  FLUIDS
arose from our assumption that sin 9 was a factor of ^, the other factor being independent of 6. This however is not the case. For, for given values of r and t, % is a finite function of 9 from 9 = 0 to 6 = tt. 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 he expanded in a convergent series of sines of 9 and its multiples; and since -% and its derivatives with respect to 9 alter continuously with d, the expansions of the derivatives will be got by direct differentiation*. This being true for all other pairs of values of r and t, % can in general be expanded in a convergent series of sines of 9 and its multiples; but the coefficients, instead of being constant, will be functions of r and t, or in the particular case of steady motion, functions of r alone. Now a very slight examination of the general equations will suffice to shew that the coefficients of the sines of the different multiples of 9 remain perfectly independent throughout the whole process, and consequently had we employed the general expansion, we should have been led to the very same conclusions which have been deduced from the assumed form of %.
47. If we take the impossibility of the existence of a limiting state of motion, which has just been established, in connexion with the results obtained in Section III., we shall be able to understand the general nature of the motion of the fluid around an infinite cylinder which is at first at rest, and is then moved on indefinitely with a uniform velocity.
The fluid being treated as incompressible, the first motion which takes place is impulsive. Since the terms depending on the internal friction will not appear in the calculation of this motion, we may employ the ordinary equations of hydrodynamics. The result, which is easily obtained, is
Vn*
Rdr + ®rd0 = d(f>, where (/> = - — cos 0f
* See a paper "On the Critical Values of the Sums of Periodic Series," Camb. Phil Trans. Vol. vm. p. 533. [Ante, Vol. i. p. 236.]
t According to these equations, the fluid flows past the surface of the cylinder with a finite velocity. At the end of the small time t' after the impact, the friction has reduced the velocity of the fluid in contact with the cylinder to that of the