RECENT PROGRESS IN HYDRODYNAMICS. 67 



bined with a motion about the axis.' ' This may be compared with 

 Boussinesq's results for a circular tube of rectangular section, referred to 

 above. 



This investigation by Stokes is a most important one, forming as it 

 does the first application of mathematical reasoning with any success to 

 the problem of the motion of pendulums in non-perfect fluids. The second 

 part of the paper, forming about one-third of the whole, is devoted to the 

 discussion of the observations of Baily, Bessel, Coulomb, and Dubuat, and 

 the values of /u for water and air are deduced from their experiments. The 

 investigations with reference to cylinders and vibrating discs have been 

 already referred to. 



Stokes had considered the case when the fluid was originally at rest 

 without vortex motion in any part, and had, therefore, taken only those 

 particular solutions of the difierential equations which i/^i, \p2 satisfy, in 

 which d enters as the factor sin ^d. In 1870, O. B. Meyer ^ took up this 

 question and showed that the general solutions are expressed in the forms 



^b^ = S(Aen + BZ„)E„ exp (XH), .//^ = 2(09^ + DZ„)Sn exp (XH) 



where 9, Z are functions of d alone, and R, S of r, both of flnite forms. 

 In fact, the 9, Z are proportional ^ to sin ddP /dd, where P is a zonal 

 spherical harmonic. The work is too complicated to be described with 

 any fulness here, but it is carried out with much skill, account being 

 taken of a concentric boundary. He proves, amongst other things, that 

 the motion due to the original state of the fluid decreases indefinitely with 

 the time, i.e., that the equation giving the values of X has all its roots 

 pure imaginaries except for the case ?i = 1 or 9; = sin^fl (Stokes' case), 

 in which it is complex, giving periodic vibrations with decreasing ampli- 

 tudes. In the particular case where the external boundary is infinitely 

 large, Meyer's results agree with those of Stokes for the time, but there 

 is a slight difference for the log decrement. When the surrounding fluid 

 is elastic the results must be modified ; this has also been considered by 

 Meyer '' who has determined the log decrement under these conditions, 

 and has shown that when the velocity of sound in the fluid is very great 

 the correction may be neglected without sensible error. Meyer refers to 

 a paper by 0. J. H. Lampe ^ on the same subject, but this I have not 

 seen. 



The investigations both of Stokes and Meyer are based on the stream 

 function, but this is only suitable when the motion takes place in planes 

 through an axis. When this is not so, recourse must be had to the 

 quaternion potential first introduced by Helmholtz ^ in his paper on 

 vortex motion. The investigation on these lines has been carried out by 



' ' On the theories of the internal friction of fluids in motion,' &c. Camh. Phil. 

 Trans., viii. ; or reprint, vol. i. p. 103, 1845. The connection of this with part of 

 Siemens' theory of the conservaiion of the sun's heat is evident. 



- ' Ueber die pendelnde Bewegnng einer Kugel unter dem Einflusse der inneren 

 Eeibung des umgebenden Jlediums,' Bmxh. Ixsiii. p. 31. 



' ' Ueber die Bewegung einer Pendelkugel in der Luft,' Borcli. Ixxv. p. 336, 

 1873. 



* Ibid. 



* ' Ueber die Bewegung einer Kugel, welche in einer reibenden Fliissiglceit um 

 einen senkrechten Durchmes.ser als feststehende Axe rotirend schwingt,' Frogramme 

 des iStddtiscIien Gymnasiums zu Danzig, 1866. 



« See Part I. of this report, Brit.- Assoc. Bep. 1881, p. 60. 



F 2 



