344 



NATURE 



[Feb. 12, 1880 



remark is called for, is the proof of the permanence of the 

 velocity potential. Mr. Lamb has offered a proof of this 

 now historic theorem, which, if judged by the space it 

 occiij ies, 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 dx, dy, ds 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. If this proof is found to be 

 unsound, it is an unnecessary blemish in the book, for 

 even if true, it would not replace the more elaborate, but 

 much more physically instructive, proofs given by Stokes 

 and Thomson, which the author has given further on in the 

 book. 



These remarks only carry us into the second chapter. 

 The rest of the book, with the exception of the last 

 chapter, is devoted to the account of what has been done 

 in the way of integrating the equations of motion, and 

 this may be taken as the purpose of the book. 



This part of the theory, which is now very extensive, 

 has almost all been developed within the last fifty years, 

 and most of it within a much shorter period. It is the 

 work of the very ablest mathematician, and is of the 

 highest and most difficult kind, and in general incom- 

 plete. I' was only to be found in isolated memoirs in 

 various languages. The collecting, abbreviating, and 

 arranging this into a systematic treatise has been no 

 ordinary task, and the result shows that, in addition to 

 his mathematical power, the author must possess the 

 gift of compiling. One of the most striking features oi 

 the book, considering the variety of sources from which 

 ' the matter was collected, is the uniformity of the nota- 

 tion. There is, however, one departure from this which 

 is important, although evidently an oversight. The term 

 stream-lines, carefully defined in Art. 28, as applicable 

 only to steady motion, is freely used throughout the book 

 in the sen^e of lines of motion, as applied to cases in 

 which the motion is not steady. 



The advance which has been made of late years has 

 not been by the discovery of any general method of in- 

 tegrating the equations of motion, but by the discovery 

 of certain general relations between the motion within 

 certain regions of space, and the shape and motion of the 

 boundaries to those regions. The steps in the discovery 

 of these kinematical relations are principally due to 

 Green, Stol es, and Helmholtz, but they have been gene- 

 ralised and elaborated by Thomson and Maxwell, and 

 to these latter the present method of expression is due. 

 An extremely lucid account of these relations is given in 

 Chapter III., by which the author has cleared his ground 

 for the treatment of such integrations as have been 

 effected. These comprise many cases of steady flow } 

 the method being that of the stream-line function first 

 given by Stokes, but afterwards reduced to a geometrical 



form by Maxwell, and largely applied by Rankine. They 

 also comprise cases of vortex motion treated by Helm- 

 holtz' s well-known method, and the theory of waves, as 

 worked out principally by Stokes, Green, and Rankine. 

 Only one chapter of the book is devoted to elastic fluids, 

 and this, under the shadow of Lord Rayleigh's complete 

 work, does not call for special comment. 



The last chapter is on viscosity, and is taken from Prof. 

 Stokes's paper on this part of the subject. Although 

 this paper has been published thirty-three years, this is the 

 first treatise in which any adequate account of its very 

 important contents has appeared in a general treatise. 



Throughout the book the various steps are carefully 

 ascribed to their different authors, a very difficult task, 

 and one in which the author appears to have been gene- 

 rally successful. There are, however, two instances of 

 failure which call for notice. Equation 10, Art. 29, is 

 known by modern French writers as Bernoulli's theorem 

 (" Thdoreme de Daniel Bernoulli, Bresse," vol. ii. p. 25). 

 Example 11, Art. 97 — The fact that the contraction from 

 a canal projecting inwards is \ was proved long ago 

 and the results verified by Borda. 



In respect of diagrams Mr. Lamb's book might certainly 

 have been improved. The great difficulty in the study of 

 the subject is to obtain a conception of the lines of 

 motion, and in this, diagrams such as those given by 

 Rankine, Maxwell, and Sir William Thomson, are in- 

 valuable. The graphic method of obtaining the lines of 

 motion developed by Maxwell and Rankine, has led to 

 most important steps, but w-ithout diagrams it is as im- 

 possible to form a conception of this method, as of the 

 lines of motion themselves. 



The omission in this respect, as well as a tendency to 

 reduce verbal explanations, would have shown without the 

 examples at the end of the volume, that the author has 

 been influenced by a desire to adapt the book to the 

 requirements of the mathematical tripos, in which desire 

 he has certainly succeeded. Whether it is well to introduce 

 students to such a difficult, complex, and uncomplete 

 subject in such a concise, not to say cut and dry form, 

 is a question which the author probably did not feel it 

 necessary to consider. He has, however, by the numerous 

 references throughout the work, and in the table of 

 authorities at the end, done all in his power to put the 

 students in the way of consulting the original works. 

 This is aid of which students will do well to avail them- 

 selves, for nothing can equal work from the master's hand, 

 and however carefully the general features may have been 

 studied, the reading of such papers as those of Stokes, 

 Rankine, and Helmholtz cannot fail to shed, what may be 

 called, the light of life over the whole subject. 



Osborne Reynolds 



THE INTERIOR OF GREENLAND 

 Meddehlser am Grbnland, udgivne af Commissions for 

 Ledehen af dc geologiske og ^eographiske I 'ndersogdser 

 i Grbnland. Fors. Hefte. (Copenhagen, 1879.) 

 CO large an amount of interest has been awakened 

 <-^ during recent years concerning the nature of the 

 interior of the vast island of Greenland that the publica- 

 tion of this first instalment of the researches carried on 

 under the auspices of the Danish Government will be 



