Feb. 12, 1880] 



NATURE 



343 



book. These remarks, although directed to the matter in 

 the book, do not, excepting one rather important case, 

 refer unfavourably to anything for which the author is 

 responsible. 



Of all subjects on which to produce a satisfactory text- 

 book, perhaps the theory of the motion of fluids, as actual 

 fluids, presents the greatest difficulties. The phenomena 

 of fluid motion, at once commonplace and very obscure, 

 have excited so little interest and called forth such slight 

 observation that at the present time a writer is unable to 

 set before his readers any adequate description of the 

 phenomena which it is his implied object to explain. And 

 as regards the theory, he has to begin by apologising for 

 his fundamental assumptions as being obviously contrary 

 to facts, and after carrying his readers through most diffi- 

 cult and complex mathematics, he has again to apologise 

 for his conclusions, which are in general contrary to ex- 

 perience. As applied to one class of phenomena— that of 

 waves— it is true the theoretical results accord closely 

 with facts ; but the satisfaction to be derived from this is 

 largely mitigated for want of a sufficient reason why the 

 theoretical conclusions should be right in this case while 

 they are entirely wrong in others, such as the flow of 

 fluids and the resistance offered to the motion of solids. 

 The usual explanation, that in the theory no account is 

 taken of the friction or viscosity of actual fluids, is hardly 

 satisfactory since no reason has been found why friction 

 should play any other part than the altogether unim- 

 portant part which it plays in the case of waves. 



It is, however, only in its application to actual fluids 

 that the theory is unsatisfactory. If it be cut adrift from 

 its origin, and be considered as a branch of abstract 

 mathematics relating only to ideal matter having the 

 properties assigned, it occupies the place of one of the 

 most advanced as well as the most important branches of 

 philosophy. It has been partially viewed in this light 

 since the middle of the last century, when Eulcr and 

 Lagrange founded the modern theory, and the tendency 

 so to regard it has greatly increased of late with the 

 development of the theory. The greatest success, indeed 

 the only real success, has been obtained by the rigorous 

 development of the theory of the motions in a perfect 

 fluid, as it is called, regardless of whether or not these 

 motions take place in actual fluids. Certain of the 

 motions are then seen to agree with the actual motions, 

 and wherever this is the case the theoretical motions 

 have taught many things about the actual motions, as, for 

 instance, the trochoidal motion of the fluid elements in a 

 wave, for which we might, otherwise, have groped for 

 ever without apprehending them. It is, however, the 

 observed motions of actual fluids which suggest the 

 problems ; and of course the greater and truer the know- 

 ledge of actual phenomena the more chance there is of 

 success in the study of the ideal fluids. But what tends to 

 retard its development and greatly to confuse the subject 

 is the mixing up, with the rigorous reasoning, of surmises 

 as to the behaviour of actual fluids, as, for example, that 

 the non-divergence of a stream of water when flowing 

 from a pipe into a large vessel is owing to an actual 

 opening having been formed in the fluid ; a surmise 

 which is at once negatived by the fact that the same 

 phenomenon occurs in the case of air in which such dis- 

 continuity is impossible. The present work is in the 



main free from such surmises, and such as there are, are 

 not the work of the author, but even these he would have 

 done well to have omitted. 



In his description of the methods by which the equa- 

 tions of motion are obtained the author has included 

 (Art. 12) a very important method first given by Maxwell, 

 which method is given at greater length in Note A at the 

 end of the volume, otherwise he has followed previous 

 writers as far back as Laplace. Considering its difficulty. 

 the fundamental reasoning is, on the whole, well put. 

 But there is a considerable amount of vagueness attend- 

 ing the author's use of the term particle. Having rightly 

 defined fluids as being such " that the properties of the 

 smallest portions into which we can conceive them divided 

 are the same as those of the substance in bulk," he pro- 

 ceeds to reason about a particle as though it were a 

 discrete quantity, the position of which is defined by some 

 point, thus ignoring the fact that, according to his defini- 

 tion, the same particle of fluid may at one time be a 

 sphere, at another a filament of indefinite length, or a 

 sheet of indefinite breadth. This vagueness appears to 

 have led him into error in Art. 1 1. 



Art. S on the equation of continuity seems to be un- 

 necessarily bare of explanation. There used to be an 

 impression that, as the name implied, the equation of 

 continuity did in some mysterious way involve the con- 

 dition that the fluid should be continuous in space. 

 Thomson and Tait, however, have in Art. 191 of their 

 volume effectually dispelled this notion. They say : — 



"As there can be neither annihilation nor generation of 

 matter in any natural motion or action, the whole quantity 

 of matter within any space at any time must be equal to 

 the quantity originally in that space, increased by the 

 whole quantity that has entered it, and diminished by the 

 whole quantity that has left it. This idea, when expressed 

 in a perfectly comprehensive manner for every portion of 

 a fluid in motion constitutes what is called the equation of 

 continuity, a needlessly confusing expression." 



The meaning of this can be nothing less than that the 

 equation of continuity has nothing to do with continuity 

 in space ; for certainly there is no creation or annihilation 

 of matter amongst the stars, probably fluids, and yet we 

 should hardly consider them continuous in space. As this 

 Art. 191 stands the last sentence is erroneous, and is cer- 

 tainly calculated to increase the confusion. To render it 

 true, the term fluid must be understood continuous fluid. In 

 deriving the equation, the constancy of mass is certainly 

 taken as an axiom, but that is not all ; when it is said that 

 the mass in a certain volume V\% pV, p is understood to 

 be the ratio of the mass to the volume in a space, so small 

 that it may be neglected as compared with V, at any point 

 within V. And hence the assumption, fundamental to the 

 equation of continuity, that the mass within V is pV is equi- 

 valent to assuming that the matter is uniformly distributed 

 through V, and therefore cannot be discontinuous. Never- 

 theless, it would have been better to have called the 

 general equation the equation of density. But it is clear 

 that this general equation was an afterthought, and that 

 the name originated from consideration of water or an 

 incompressible fluid, in the case of which the equation 

 does not involve the density, and simply expresses space 

 continuity within a substance of constant volume. 



Another point of fundamental importance, on which a 



