June 17, 18S0] 



NA TURE 



145 



the terms cirro-stratus and strato-cirnis. Finally the terms 

 cuintih-stratiis and cirro-stratus are sorely needed for varieties of 

 clouds intermediate between the class I have described and the 

 cuinubis and cirrus types, if any part of Howard's terminology 

 is to be left to us at all. 



It would be a pity that that terminology, lucid and expressive, 

 should perish, merely because, to a few minds, the originator 

 of a system must needs appear infallible, and his classification 

 perfect as Minerva when issuing from the head of Jupiter. I 

 think that Luke Howard would have been the last to put forward 

 such a claim. W. Clement Ley 



Junes 



The Motion of Fluids 



Prof. Reynolds, in the course of his review (Nature, vol. 

 xxi. p. 342) of my book on the above subject, cites two instances 

 in which I have been guilty of what he considers loose and vague 

 reasoning. I would ask space for a few remarks on the points 

 in question. 



To take the more important matter first. Prof. Keynolds says, 

 apropos of a certain proof of the velocity-potential theorem 

 given in Art. 23 : — 



" Mr. Lamb has offered a proof of this now historic theorem, 

 which, if judged by the space it occupies, should be much simpler 

 than the acknowledged proofs of Cauchy and Stokes. As no 

 authority is cited, it would appear that this proof is here 

 given for the first time. If so, the author has done himself great 

 injustice in not examining or explaining his reasoning more 

 closely. For, as it stands, it suggests the idea that he has ignored 

 the fact that d x, <iy, r/ ;, on the left of his equation, are integrals 

 through a finite time, and hence, inasmuch as he has given no 

 reason to the contrary, may be of a different order of magnitude 

 from^ their initial values, da, db, dc, which appear on the right 

 of his equation. If this is not so it is a peculiarity of the motion 

 of continuous fluid, and needs establishing ; otherwise we might 

 infer that two people who had once shaken hands could never 

 after be so much as a mile apart." 



Prof. Reynolds, who himself strongly recommends the careful 

 study of "work from the master's hand," will hardly take it 

 amiss if I ask him to turn to the proofs which he justly cites as 

 classical, and to notice that tliey contain, one of them (Cauchy's) 

 in exactly the same form, the other in a form which is mathe- 

 matically equivalent, the very assumption ^\hich he here calls in 

 question. The assumption is in fact nothingmorethan a tacit limita- 

 tion, which is made at the very outset of the subject, as to the class 

 of motions which are proposed for study. In the " Eulerian " 

 method it is implied that the first derivatives of the component 

 velocities u, v, ot with respect to the co-ordinates .x,y, z are to be 

 ever>^vhere and always finite throughout the motion considered ; 

 in the "Lagrangian" method the corresponding, and equivalent, 



ossumption is that the derivatives ^-^, — , ^, &c , and also 



da d b d c 

 d- X d- X d- X ^ , . . „, , 



TTjt' jTZr,' J j^ ' ^'^•> ^i^fi to te finite. We do not assert 

 aadt dbdt dcdt 



that these are universal characteristics of fluid motion, for it is 

 easy to imagine cases in which they are violated ; we merely 

 exclude such cases ab initio from the scope of our investigations. 

 But, in one form or another, these fundamental limitations are, 

 from the point of view of analytical hydrodynamics, unavoid- 

 able ; they are made implicitly every time we write down the 

 equations of motion, and it is therefore not surprising that they 

 should be found to be essentially involve,!, not only in the proof 

 which Prof. Reynolds on this account criticises, but in every 

 other proof of the velocity-potential theorem which has yet been 

 propounded. 



I have only to add that the proof in question is, and professes 

 to be, merely a very obvious corollary to H. Weber's transfor- 

 mation of the Lagrangian equations. 



The other passage of Prof. Reynolds's review which I wish to 

 notice is as fellows : — 



" There is a considerable amount of vagueness attendm^; 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 proceeds 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 definition, 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. 11. "| 



A good deal of this criticism is, I think, met by the remarks 

 already made. In a fluid moving subject to the conditions I 

 have stated, only finite changes of shape can be produced in a 

 moving element within a finite time. 



Prof. Reynolds does not indicate the precise nature of the 

 "error" which he finds in Art. 11. Aftera careful reconsidera- 

 tion, the argument of that article appears to me to be sound ; but 

 I am free to confess that it is not stated with all the clearness 

 desirable, and that the article is further disfigured by an un- 

 fortunate clerical error in the foot-note, where ";< = ± sj x" 

 should be read for "u ='±.v." Horace Lamb 



Adelaide, March 30 



On the Physical Aspects of the Vortex-Atom Theory 



Will any charitable person explain a difficulty which I (and 

 other non-mathematical people) have encountered when seeking 

 to understand and be satisfied with this theory ? 



The only proof of those properties of vortex rings which match 

 the physical properties of atoms that I have met with is that in 

 Besant's "Hydromechanics"; and is based on the initial-co- 

 ordinate method. 



Now it seems to me that this method assumes what is equiva- 

 lent to the permanence of the vortex filament ; so that in proving 

 the latter by use of this system of co-ordinates we may t>e merely 

 arguing in a circle. 



For it assumes that if initially we have any infinitesimal tetra- 

 hedron 5 a . 5 ;8 . S 7, then after the finite time, t, this will still 

 form a tetrahedron 5. v. 5/ . 3:, 



Now I cannot see that one can assume this ; that — to use the 

 words in a late article of Nature — "if two people have once 

 shaken hands they can never be too miles apart." 



And this inseparability of the particles of a fluid thus assumed 

 bears a very close relation to the permanence of the vortex 

 filament which we wish to prove. W. L. 



Cheltenham, May 29 



[It appears to us that our correspondent here confuses between 

 the permanence of any fluid filament and the permanence of the 

 vortex character of the filament. The assumption that every 

 filament remains continuous cannot be said to be equivalent to 

 assuming that the direction of the filament at every point remains 

 coincident with the axis of rotation of its constituent elements at 

 that point, which is what Helmholtz has taught us. — Ed.] 



The Aurora Borealis and its Colours 

 With regard to Drs. De La Rue and Midler's paper on the 

 Aurora (Nature, vol. xxii. p. 33) there is still a point I should 

 like to see explained. Is it considered by physicists that in 

 electric discharges similarity of colour is sufficient to indicate 

 similarity of constitution, even when their spectra are quite 

 unlike ? The paper, together with the reply to Prof. Smyth, 

 certainly seenn to imply this ; though I have not previously seen 

 it stated to be the case. 



With regard to the red part of aurorce, so far as my observa- 

 tions indicate its position, they show it to be above the greenish 

 part in the aurora; seen here ; though according to Weyprecht's 

 observations, it is below the green in the Arctic regions. 

 Sunderland, June 9 T. W. Backhouse 



A New Audiphone 



Further experiments on t'ae timbre of musical instruments 

 as rendered by the audiphone have led me to the selection of the 

 following as a distinct improvement on the birchwood veneer, 

 both for musical purposes and also for ordinary conversation. 

 It has the same advantage as my previous form in not requiring 

 to be held by the hand, it costs nothing, and requires no making. 

 Takeasheet of stiff brown paper about II x 15 inches, the paper 

 being such as is ordinarily used for making up heavy parcels. 

 Put the ends together, the middle forming a loop, and hold the 

 ends between the teeth. The paper must be pretty stiff, as the 

 loop must stand out round and full, and of course the paper must 

 be without folds or creases. Thos. Fletcher 



Museum Street, Warrington 



Crystal-Ice 



In reference to the "crystal ice" proposed by Dr. Calan- 

 tarients, of Scarborough, for skating upon with ordinary skates, 



