March 30, 1893] 



NATURE 



513 



to be solved by fairly simple methods. There is, however, 

 another class of problems of great practical importance, in which 

 it is not allowable to neglect these quadratic terms, and towards 

 the solution of such problems theory has as yet made little 

 progress. 



When a sphere is constrained to move along a horizontal 

 straight line, but is otherwise free, it is well known that if the 

 surrounding liquid is supposed to be frictionless, its only effect 

 is to increase the inertia of the sphere by half the mass of the 

 liquid displaced. The sphere accordingly requires a larger 

 impulsive force to start i: than if the liquid were absent, but 

 when once started it continues to move with its velocity of pro- 

 jection. But when the sphere is surrounded by an actual 

 liquid, its velocity gradually diminishes until it ultimately comes 

 to rest ; and this fact shows very forcibly the necessity of taking 

 the viscosity of the liquid into account in problems of this cha- 

 racter. I obtained a few years ago a mathematical solution, 

 which shows that this effect must necessarily be produced by a 

 viscous liquid, but the solution is an imperfect one, as mathe- 

 matical difficulties compelled me to disregard the quadratic 

 terms. 



It is always a great advantage when the solution of a mathe- 

 matical problem can be made to depend upon a single function 

 which satisfies a partial differential equation and certain boundary 

 conditions. This is always the case when a solid of revolution 

 moves along its axis in a viscous liquid which is initially at rest, 

 or has an independent motion which is symmetrical with respect 

 to the axis. In this particular class of problems, the motion can 

 be expressed by means of Stokes's current function in the follow- 

 ing manner:— Let z be measured along, and r perpendicularly 

 to the fixed straight line with which the axis coincides during 

 the motion ; let w and ti be the velocities of the liquid in these 

 directions ; then : — 



u= - ~ -T, 7t' = - -f, 

 r dz r dr 



D = 



dr 



dz r I 



dr' 



and V is the kinematic coefficient of viscosity. 



So far as I am aware, no serious attempt has been made to 

 obtain a solution of this equation in a suitable form, even when 

 the solid is a sphere. The equation is well worthy of the 

 attentive consideration of mathematicians ; and although it is an 

 intractable one, it must be recollected that a general solution is 

 not required, but only a particular one which is suitable in the 

 case of a sphere. It will be quite time enough to consider the 

 possibility of obtaining solutions of a more general character, 

 when the appropriate one in the case of a sphere has been dis- 

 covered. It is also important to recollect that in most problems 

 which are of practical interest, «/ is a small quantity (about "014 

 in C.G.S. units for water), and consequently an approximate 

 solution in which v is supposed to be small would meet the 

 exigences of the case. 



When a solid body is moving through a liquid, one of the 

 boundary conditions is that the normal velocity of the solid 

 must he equal to the component along the normal of the velocity 

 of the liquid in contact with it. If the liquid is frictionless, this 

 condition is the only one which has to be satisfied ; but when 

 the liquid is viscous, a further question arises as to the law which 

 expresses the effect of the tangential stress exerted by the liquid 

 upon the solid. When the motion is very slow (as in the case 

 of problems relating to small oscillations) the experimental 

 evidence is in favour of the hypothesis of no slipping ; but when 

 the velocity is considerable, the experimental evidence is not so 

 satisfactory. The partial slipping which takes place under these 

 circumstances must depend partly upon the nature of the liquid, 

 and partly upon that of the surface in contact with it ; and the 

 tangential stress to which it gives rise is probably approximately 

 proportional to the square of the relative velocity. 



When the motion is symmetrical with respect to an axis, 

 the stresses due to viscosity can be calculated as soon as the 

 value of i^/ is known, the resistance which the liquid exerts on 

 the solid can be found, and the equation of motion written down 

 and integrated. This process is, however, an exceedingly tedious 

 one ; but it can always be dispensed with in the case of a single 

 solid by employing the principle of momentum. When the 



NO. 1222, VOL. 47] 



motion IS not symmetrical with respect to an axis, it cannot be 

 expressed in terms of if> ; but if the velocities of the liquid can 

 be found from the hydrodynamical equations, the components of 

 the linear and angular momenta of the liquid can be calculated, 

 and by applying the principle of momentum to the compound 

 system composed of the solid and the surrounding liquid, the 

 equations of motion of the former can be obtained. Since the 

 momentum of the system is obviously a function of the six co- 

 ordinates of the solid, this principle furnishes a sufficient number 

 of equations for the determination of the motion. 



When there is more than one solid, the principle of momentum 

 is insufficient to determine the motion ; but if the velocities of 

 the liquid in the neighbourhood of each solid could be found, 

 the force and couple constituents of the resistance could be 

 calculated, and the equations of motion of each solid written 

 down. Lagrange's equations in their ordinary form cannot be 

 employed, as viscous motion involves a conversion of energy 

 into heat ; but problems which can be solved by an indirect 

 method can usually be solved by a direct one, and I feel con- 

 fident that equations analogous to Lagrange's equations exist, 

 by means of which the motion of a number of solids in a viscous 

 liquid can be found without going through the above-mentioned 

 process. A form of Lagrange's equations has already been dis- 

 covered, which is applicable when the viscous forces depend 

 upon a dissipation function which is expressible as a homogeneous 

 quadratic function of the velocities ; and the circumstance that 

 a dissipation function also exists in the hydrodynamical theory, 

 although it is expressed in a different form, furnishes additional 

 grounds for believing in the existence of equations of this cha- 

 racter. The discovery of such equations would constitute an 

 important advance in the theory of viscous liquids, 



A. B. Basset. 



SCIENCE IN THE PUBLIC SCHOOLS AND 

 IN THE SCIENTIFIC BRANCHES OF THE 

 ARMY. 



ON Friday last Mr. Campbell Bannerman received a 

 deputation on this subject in his room in the House 

 of Commons. There were present Sir Henry lloscoe, 

 the Head Master of Rugby School, the Principal of 

 Cheltenham College, the Head Master of Clifton College, 

 Sir B. Samuelson, Prof. Jelf, and Mr. Shenstone. Lord 

 Playfair, Sir John Lubbock, and Sir Henry Howorth 

 would also have been present, but they were prevented by- 

 other engagements. The following is a brief account of 

 the proceedings : — 



Sir Henry Roscoe, in introducing the deputation, said that 

 he had introduced a deputation on this subject to Mr. Stanhope 

 about five years ago, and that if the suggestions then made had 

 been adopted the present deputation would not have been 

 necessary. After some remarks which showed the injustice of 

 the present system to the more scientific lads, he pointed out 

 several methods by which this injustice might be removed. 



The Head Master of Rugby, Dr. Tercival, expressed his 

 strong feeling of the importance of the subject alike to the 

 service, the cadets, and the schools, and said he wished to see 

 both modern languages and science duly encouraged; he 

 thought they might both be made compulsory, as he believed 

 that early education should rest on a wide basis, and that 

 specialising should only be encouraged later. Alluding to the 

 work in science done at the Royal Military Academy, Dr. 

 Percival mentioned that he knew of one cadet who, owing 

 to the absence of any higher teaching there at the earlier 

 stages, was lately learning science which he, the cadet, was 

 well fitted to teach. 



The Principal of Cheltenham College, Mr. James, confessed 

 that his own interests and convictions on educational matters 

 were those of a linguist rather than those of a man of science ; 

 but practical experience showed him that the present system 

 told most unfairly against scientific boys who entered Woolwich ; 

 science was being gradually edged out. Many other head 

 masters of public schools felt with the deputation. He thought 

 also that the present system tended to the disadvantage of the 

 smaller schools, where science was often exceedingly well 

 taught. He hoped that in making any changes the authorities 

 would be careful to consider the interests of linguistic boys, and 



