200 



NATURE 



\yan. I I, 1872 



finite terms several most important partial differential 

 equations of the second order, including the equation of 

 continuity in a homogeneous incompressible fluid ; and 

 the chapters in which these equations are discussed are by 

 far the most important and interesting in the work. Mr. 

 Earnshaw is already known for his able treatment of the 

 equation for the motion of a sound wave in the Philoso- 

 phical Transactions for 1 860, and no one can doubt the 

 importance of the subjects suggested for consideration by 

 this and other equations. The question is discussed 

 whether there must necessarily e.xist an integral of every 

 partial differential equation that can be proposed, and on 

 this part of the subject we wish the author had extended 

 his remarks. The real question considered seems how- 

 ever rather to be the possibility of the existence of 

 a continuous function expressible in finite terms as an 

 integral. With regard to the considerations having 

 reference to certain physical problems, we should not 

 expect to learn very much from the discussion of such 

 questions, as the differential equation might admit of 

 a solution incapable of satisfying the physical conditions. 



We must notice one singular error mide by Mr. 

 Earnshaw. He concludes that the well-known partial 

 differential equation of the second order of surfaces having 

 their principal radii of curvature equal and of opposite 

 signs at all points, admits of no integral, because the 

 form of a surface possessing this property would be such 

 as could not exist ; but it is well known that the surface 

 formed by the revolution of a catenary round its directrix 

 does possess the property in question, and it is easy to see 

 that this arises simply from the fact that the normal and 

 radius of curvature in the catenary are equal and of 

 opposite signs ; the form of the surface is quite easy 

 to conceive. A particular integral of the equation ob- 

 tained by Poisson's method is also given in Boole's 

 Differential Equations, chapter xv. Even admitting Mr. 

 Earnshaw's reasoning, it would only establish the non- 

 existence of a real surface possessing the required property. 

 The integrals of the equation of continuity in three 

 dimensions, and of one or two other equally important 

 equations, we do not remember to have seen before, and 

 they are perhaps the most general finite solutions the 

 equations admit of Of the value and power of the 

 method it is impossible to speak at present ; but we 

 heartily commend Mr. Earnshaw's book to the reader as 

 one containing much matter of great interest systemati- 

 cally and clearly developed and treated by a novel method. 

 It is remarkable that the subject of partial differential 

 equations has not attracted more attention than it has in 

 recent years, as an advance in this quarter is more imme- 

 diately felt in physics than an advance in any other pure 

 mathematical subject. The present work will help to 

 bring the matter prominently forward ; and as the analysis 

 is nowhere of a very difficult nature, it will probably come 

 under the notice of many readers not accustomed to study 

 mathematical memoirs on their appearance. 



If the work had been intended to be a Treatise on the 

 subject, we should have had good reason to object to the 

 total omission of all reference to the usual methods, but 

 the title and preface explain that this was not contem- 

 plated ; it is one of the few English books containing 

 original mathematics. 



J. W. L. G. 



OUR BOOK SHELF 



Three and Four Place Tables of Logarithmic and Trigo- 

 nometric Functions. By James Mills Peirce. 16 pp. 

 (Boston : Ginn Brothers, 1871.) 

 Perhaps the best way of treating this work, which 

 does not contain a single word of explanation, will be to 

 give a summary of the tables contained in it. First we 

 have proportional parts of all numbers up to 100 ; then 

 on one page three-place logarithms of numbers and of the 

 six trigonometric functions, natural and logarithmic. On 

 pages 4 and 5 we find four-place logarithms of numbers, 

 then logarithms of sums and differences (Gaussian loga- 

 rithms) also to four places, then follow tables of logarith- 

 mic trigonometric functions, inversetrigonometric functions 

 (a new table, to which attention is specially invited, for 

 finding angles from the logarithms of their trigonometric 

 functions), traverse table, the correction of the middle 

 latitude (in an improved form), and meridional parts. 



In a prospectus issued by the publishers, it is stated as a 

 result of experiment that it has been found that the times 

 occupied, in regular computation, in doing one piece of 

 work by tables of 4, 5, 6, and 7 places, are proportional to 

 the numbers i, 2,3, and 4 ; hence it is that the author has 

 drawn up the majority of the tables under review to 4 

 places as sufficient for ensuring the degree of accuracy 

 usually required in computations of common surveying, 

 engineering, &lc. 



The type employed is very clear, the arrangement of 

 the work is good, and the printer's part has been well 

 done ; the book requires only a few words of elucidatory 

 matter. There is on the last page a useful Table of Con- 

 stants with their logarithms, here we observe a few symbols 

 which are new to us, and which are presented to our 

 notice on the Title-page. 



After all the value of such a work consists in its accu- 

 racy, and that can only be tested by practice, " the 

 greatest pains have been taken both in preparing and 

 printing to secure perfect accuracy." We commend the 

 work to the notice of such as agree with old Burton 

 (Anatomy of Melancholy, pt. 11., sec. 2), "What so pleasing 

 can there be ... if a man be more mathematically 

 given (as) to calculate or peruse Napier's logarithms, or 

 those tables of artificial sines and tangents, not long since 

 set out by .... Edmund Gunter, which will perform 

 that by addition and subtraction only, which heretofore 

 Regiomontanus' tables did by multiplication and division." 

 But then the same quaint writer advises those who are 

 melancholy to square a circle ; does it follow that all circle- 

 squarers are melancholy .' R. T. 



The Laws of the Winds prevailing in Western Europe. 



By W. Clement Ley. Part I. (Stanford, 1872.) 

 Even when we differ from an author's conclusions, the 

 work of one who shows himself an honest and capable 

 inquirer has a just claim to our attention. Mr. Ley 

 evidently writes from practical knowledge of his subject, 

 and his assiduity in collecting and charting observations 

 must have entailed on him an amount of labour which 

 only those who have been engaged in similar work can 

 thoroughly understand. Unfortunately, as it appears to 

 us, he has confined his investigations almost entirely to 

 the limits set forth on his title-page ; and the winds of 

 Western Europe, though highly suggestive and subject to 

 more exact observation than any others except those of 

 the United States, are by no means to be taken as repre- 

 sentative. Mr. Ley has taken them as such, and has thus 

 laid down a series of general propositions, which may be 

 briefly summed up in one — that revolving storms are 

 caused by the barometric depression consequent on heavy 

 rain over a large area. He brings forward some curious 

 home instances in illustration of this ; but looking farther 

 afield, on the slopes of the Himalayas — to mention only 

 one locality — a much heavier and longer continued pre- 



