April 26, 1894J 



NATURE 



599 



that results have been demonstrated which have really 

 only been suggested as possibly true. A case in which 

 the criterion we have laid down is not satisfied, occurs 

 on p. 35, where, after having shown that a finite series of 

 sines can be found, the sum of which coincides with the 

 values of a prescribed function at n points, the author 

 states that since this result holds good however large n 

 may be, the limiting form of the curve represented by the 

 series absolutely coincides with the arbitrarily given 

 curve between the limits of the variable. A precisely 

 parallel argument would show that a similar result was 

 true for a power series, which is well known not to be the 

 case. No sufficient safeguard is given by the statement 

 on the next page that the infinite series must be con- 

 vergent, or by the limitation introduced on p. 38. The 

 method by which Fourier's double integral is obtained on 

 p. 53, is another example of a case in which the student 

 will be apt to believe that the result has been proved. 

 We think that it is very doubtful whether the simplifica- 

 tion of Dirichlet's proof of the convergence of Fourier's 

 series obtained by considering a particular case of the 

 series, as in chapter iii., is sufficiently great to compen- 

 sate for the loss of generality. 



In chapter iv. a number of interesting and instructive 

 special problems in heat and in vibrations are considered, 

 a considerable number of exercises being left for the 

 student to solve. 



The treatment of spherical harmonics in chapters v. 

 and vi. is satisfactory ; a little more space might, how- 

 ever, have been with advantage devoted to the discussion 

 of solid harmonics as developed by Thomson and Tait, 

 and by Maxwell. 



In chapter vii., in which Bessel's functions are con- 

 sidered, the infinities of the two Bessel's functions, both 

 for real and imaginary arguments, should have been 

 evaluated, as the selection of the proper forms for the 

 solution of potential problems requires a knowledge of 

 the values of the functions when the argument is infinite. 

 Chapter viii. gives a good introduction to Lame's func- 

 tions, the toroidal functions being also briefly mentioned. 

 The interesting historical summary, added by Dr. Bocher, 

 adds considerably to the value of the book. 



In spite of some defects, the treatise is in many ways 

 in advance of any other on the same subjects, in the 

 English language, and should be consulted by all students 

 of mathematical physics. E. W. H. 



OUR BOOK SHELF. 



Bird Life in Arctic Norway. By Robert Collett, Pro- 

 fessor of Zoology in the University of Christiania. 

 Translated by A. H. Cocks, M.A. (London : R. H. 

 Porter, 1894. 



The snow-covered peaks of the Land of the Midnight 

 Sun possess irresistible powers of attraction for most 

 lovers of nature. And they who make periodical migra- 

 tions to this Switzerland of North Europe, as well as casual 

 tourists, cannot do better than provide themselves with 

 a copy of the popular brochure now before us. In it 

 the traits of the bird-life in the three natural zones 

 of which Arctic Norway consists will be found interest- 

 ingly treated. These three natural divisions are (i) the 

 coast district and the belt of islands girding the co-ist up 

 to North Cape ; (2) the deep fjords of the Arctic Ocean 

 and the adjacent river basins m East Finmarken ; and 



NO. 1278, VOL. 49] 



(3) the interior plateaus of Finmarken, or Lapland 

 proper. Each region is brightly described, and the 

 peculiar characteristics of the bird-life in it are plainly 

 set forth. The information imparted by the guide 

 is accurate and well adapted for the general reader ; and 

 the ornithologist will also find in it much that is worth 

 reading, especially as a list of the birds of Norway, 

 arranged according to the rules of the British Ornitho- 

 logists' Union, is given in an appendix. It would be an 

 advantage if, in future editions of the book, the names of 

 places referred to by means of capitals and dashes, thus, 



M , T , &c. were printed in full. To guess the 



locality from these designations is sometimes difficult, 

 and the signs themselves are always tantalising. 



A Text-Book of Euclid's Elements. (Books ii. and iii.) 

 By H. S. Hall, M.A., and F. H. Stevens, M.A. 

 (London : Macmillan and Co., 1894.) 



In this work the authors deal exclusively with the second 

 and third books of Euclid. Thepropositionsandtheirproofs 

 are clearly stated and proved, and very little additional 

 matter,with the exception of corresponding algebraical for- 

 mulse and exercises, is inserted between the propositions 

 themselves. Later in the book, following a few words on 

 the method of limits as applied to tangency, several of 

 the well-known theorems on Book iii., with numerous 

 examples, are given ; thus one is brought into contact 

 with problems on tangency, orthogonal circles, pro- 

 perties of the pedal triangle loci, maxima and minima, 

 iS:c., concluding with a series of harder miscellaneous 

 examples. A short appendix contains one or two pro- 

 positions on the pole and polar, and radical axes. The 

 book is thoroughly suited for work in schools and col- 

 leges, and is printed neatly with distinct figures. W. 



LETTERS TO THE EDITOR. 



\_The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part ti/"NATURK. 

 No notice is taken of anonymous communications.^ 



Panmixia. 



It is now twenty years ago that I published in these columns 

 the doctrine of Panmixia, or Cessation of Selection, and since 

 this doctrine was independently re-enunciated by Prof. Weismann 

 I have repeatedly had occasion both to explain and to defend 

 our common views upon the sabject. For it is surprising how 

 many of our foremost English evolutionists seem to have 

 found a difficulty in understandmsj exactly what is meant by the 

 doctrine. In view, therefore, of Prof. Weismann's forthcoming 

 lecture at Oxford, it seems desirable that the present standing of 

 the matter should be presented to the consideration of English 

 biologists. An opportunity may thus be afforded him of 

 answering the objections which they have raised against one of 

 the fumiamental doctrines of his entire system. 



In Nature of April 12 Mr. Wallace writes : — "He (Mr. 

 Kidd) is under the mitaken impression that the theory 

 {i.e. the state) of panmixia leads to continuous and unlimited 

 degeneration. Many writers have pointed out that this is an 

 error. The amount of degeneration thus produced would be 

 limited to that of the average of those born during the preceding 

 generations in place of the average of those that had siavived. 

 As Prof. Lloyd Morgan puts it, the survival-mean would /all 

 back to the birth-mean." 



This way of putting it, however, was originally due to Prof. 

 Ray Lankester, whose views and terms relating to the subject 

 were afterwards adopted provisionally by Prof. Lloyd Morgan. 

 It may still be remembered by your biological readers that 

 about four years ago Prof. Ray Lankester somewhat vigorously 

 attacked my views on the Cessation of Selection as a cause of 

 degeneration, and disputed their identity wiih ihose of Prof. 

 Weismann on Panmixia. He urged that by Panmixia Weis- 

 mann meant, not the merely passive cessation of selection, but 

 an active reversal of it, through Economy of Nutrition, &c. 



