196 



NATURE 



[December 29, 1904 



many have been discovered and made known by the 

 authors. Cooke's " British Desmids," issued in 

 1886-7 as a compilation of all the forms then known, 

 included less than 300 species and less than 50 varie- 

 ties. In this first volume rather more than one-fifth 

 of the British species and varieties are included, so 

 that the " Monograph " will probably extend to five 

 volumes. 



Each form is described, with references to its 

 synonyms and its bibliography ; and its distribution in 

 the British Islands is detailed, the authority for each 

 locality being stated. The figures are original, except 

 where it was not possible to procure specimens. When 

 borrowed the sources are always acknowledged. A 

 very full list of books and papers on desmids adds to 

 the value of the work. 



The " Monograph of British Desmidiaceae " is 

 worthy of a place among the numerous valuable works 

 issued by the Ray Society, and will be indispensable 

 in the study of these plants. 



THEORY OF RAPID MOTION IN A COM- 

 PRESSIBLE FLUID. 

 Legons sur la Propagation des Ondes et les Equations 

 de r Hydrodynatnique. By Jacques Hadamard. 

 Pp. xiii + 375. (Paris: Hermann, 1903.) Price 18 

 francs. 



THE theory of fluid motion, as ordinarily worked 

 out, presents several lacunae. One notable 

 omission is the absence of any detailed dis- 

 cussion of the effects of compression and rarefaction 

 of air owing to the rapid motion of bodies through 

 it. .An artillerist, seeking by the aid of the theory 

 for principles that would help him to understand the 

 resistance of the air to the motion of projectiles, would 

 be likely to be disappointed. He would find an 

 explanation of the effect of rifling in keeping the 

 points of projectiles forward ; but, while he might 

 admire the ingenuity displayed in the development 

 of the theory, he would feel that, with this exception, it 

 shed but little light upon his business. The present 

 book represents the outcome of efforts made in recent 

 years by some French mathematicians, and especially 

 by Hugoniot and P. Duhem, to widen the scope of 

 the traditional hydrodynamics so as to include rapid 

 motions in compressible fluids. 



Our hypothetical artillerist would need to exercise 

 much patience in order to get on with the book. He 

 would probably soon give it up as too intensely mathe- 

 matical. The first chapter is devoted to an account of 

 an existence theorem in the theory of potential. It is 

 to be proved that, provided a certain condition is 

 satisfied, there exists a function which is harmonic in 

 a given region and has a given normal rate of vari- 

 ation at the boundary of the region, in other words, 

 that irrotational motion of incompressible fluid is 

 possible within a closed surface which changes its form 

 in a prescribed manner without changing its volume. 

 The author gives a proof which is very interesting from 

 the point of view of analysis. He also expresses the 

 required function by means of a subsidiary function 

 which he calls " Fonction de Franz Neumann," and 

 of another which he calls " Fonction de Klein." The 

 NO. 1835, VOL. 71] 



latter is the velocity potential due to a source and a 

 sink within the given surface, and the former also can 

 be interpreted physically, but the interpretations are 

 not recorded. In the case of a spherical boundary, 

 which is worked out, the results are attributed to 

 Bjerknes and Beltrami. It would seem that these 

 writers, therefore, virtually anticipated Hicks's dis- 

 covery of the image of a source with respect to a 

 sphere. One misses the interpretation in terms of 

 images. The mathematics is there, but the author 

 does not tell us what it means. Nevertheless the 

 mathematics is excellent. 



In chapters ii. and iii. we have so much of the 

 ordinary theory as is requisite for the purpose of 

 setting out the equations and conditions which govern 

 the motions of fluids, and we have also an extension to 

 discontinuous motions. The fact that was emphasised 

 by Hugoniot is that the motion is not necessarily 

 continuous. He paid especial attention to the case 

 in which the velocity is everywhere continuous, but 

 the differential coefficients of the components of 

 velocity are discontinuous at a moving surface. The 

 discontinuities at such a surface are not arbitrary, but 

 are subject to three sorts of conditions. The surface 

 moves through the fluid like a wave. One set 

 of conditions connects the discontinuities with the 

 direction of the normal to the surface. A second set 

 connects them with the velocity of propagation. These 

 two sets of conditions are kinematical. To determine 

 the velocity of propagation the dynamical equations 

 must be introduced. The kinematical conditions are 

 called " conditions d'identit6 " and " conditions de 

 compatibility," and they are expressed by means of 

 some elegant geometry. The necessity for such con- 

 ditions has been recognised by other writers in the 

 case of discontinuities that affect the velocity. The 

 latter are here called " waves of the first order." 

 The origin of Hugoniot's discontinuities, called 

 " waves of the second order," is found in an analytical 

 paradox. If the pressure is a function of the density, 

 the equations of motion determine the acceleration of 

 every particle ; but, if the motion of a boundary is 

 prescribed, the normal component of the acceleration 

 of the particles that are in contact with the boundary 

 is prescribed also. The two values thus obtained for 

 this acceleration are in general different. Waves of 

 the second order originate at the boundary, and are 

 propagated through the fluid. 



Chapter iv. deals with rectilinear motion in a gas, 

 and is mainly occupied with the problem, first attacked 

 by Riemann, of discontinuities that affect the velocity. 

 Riemann's theory was condemned by Lord Rayleigh 

 on the ground that it violated the principle of energy, 

 and the problem remained in an unsatisfactory state 

 for many years. It was taken up again by Hugoniot 

 in 1887 without knowledge of Riemann's work. 

 Hugoniot introduced expressly the condition that the 

 increment of energy — kinetic and internal — of the 

 portion of fluid which undergoes a sudden change of 

 state is equal to I he work done upon it by the pres- 

 sures of neighbouring portions, and he concluded that 

 the law connecting pressure and density {p = Kpy) 

 cannot bo maintained during the passage of the dis- 





