264 



NATURE 



[December 6, 1917 



chosen as to give the smallest possible number of 

 changes. A little rejection shows that this will 

 always be the case," etc. The sentence we have 

 italicised contains the fault referred to. The pre- 

 sent reviewer learnt the theory of equations mainly 

 from Todhunter's treatise ; the immortal Isaac, in 

 his old-fashioned, unemotional way, does not 

 appeal to his readers' powers of reflection, but 

 does his best to show that in his diagram, and 

 any such, the last row of signs must have at least 

 one more variation than the first. We doubt 

 whether anybody could write, in a reasonable 

 space, a better explanation than Todhunter's ; 

 nevertheless, it took us a good deal of reflection 

 to appreciate it. A still more striking instance is 

 in the discussion of Sturm's theorem (p. 219). 

 Kfere we read : "It will be seen," etc., followed by 

 a statement of the theorem for a particular case. 

 We learnt Sturm's theorem in the first instance 

 from De Morgan's article in the " Penny Cyclo- 

 paedia." The great Augustus does not say: "It 

 will be seen ..." (Did he ever say so, in this 

 kind of way?) 



One more example, of a rather different kind. 

 Pp. xi-xiv contain a list of formulae, etc., which 

 the reader is supposed to know, and are given for 

 reference. Under "Binomial Theorem" we have: 

 *' Key number of term. The number of factors in 

 the numerator of any term, the number whose 

 factorial occurs in the denominator, the exponent 

 of X, and the number subtracted from m to form 

 the exponent of a are always the same number, 

 viz. n- I." Doubtless this would be lucid to the 

 late Henry James, but it -is not so to us, and we 

 do not believe that it would be so to an average 

 English student, except after a good deal of 

 previous explanation. 



Of actual mistakes we have found very few. 

 P. 20 (top) the reasoning is so vague that a student 

 might fairly argue that the proper formula is 

 n! -r! instead of n/-fr/; p. 23, in England, if the 

 probability of an event is 3/7, we say that the odds 

 are 4 to 3 against it, but "odds" may have a 

 different meaning in the States; p. 56, "a vari- 

 able can have only one limit" is wrong as it 

 stands ; p. 79 (bottom) gives a very cryptic rig- 

 marole for differentiating tt« ; p. 108, for " a 

 number" read "a fixed number," otherwise the 

 whole argument breaks down; p. 115, the expan- 

 sion of {2 - T,x + ^.x^)/(i - 2,x + 2X^) should be 

 done by synthetic division, not by undetermined 

 coefficients; p. 149, the notes about Napier's 

 logarithms are incorrect (in particular, Napier's 

 logarithms are not " natural " logarithms) ; p. 169, 

 "t represents the positive square root of - i " is 

 meaningless,, especially the "positive." 



(2) Prof. Atma Ram's book ought to be 

 extremely useful to those who can use a collection 

 of solved examples in the proper way. It is a 

 sort of abbreviated "Walton" fairly brought 

 up to date, the range being from elementary 

 dynamics and kinematics to central forces, includ- 

 ing planetary motion. So far as we have been 

 able to test them, the solutions are all correct, 

 sufficiently detailed, and often very elegant. The 

 NO. 2510, VOL. 100] 



English is thoroughly idiomatic, and Prof. Ram 

 is his own printer and publisher. Paper and 

 typography are as good as many Indian Govern- 

 ment samples; we wish that the quality could be 

 improved all round. G. B. M. 



BIOLOGY OF WATER SUPPLIES. 

 The Biology of Waterworks. By R. Kirkpatrick. 

 (British Museum (Natural History) Economic 

 Series, No. 7.) Pp. 58. (London: Printed by 

 order of the Trustees of the British Museum^ 

 1917.) Price 15. 



SINCE men of science became more intimately 

 associated with engineers in the management 

 of waterworks, questions of animal and plant life 

 in water supplies have been brought more into the 

 foreground, and it is with the object of directing 

 attention to the importance of these questions that 

 the trustees of the British Museum have placed 

 an exhibition in the South Kensington Museum 

 and have published this pamphlet as a guide 

 thereto. 



The first section, dealing with the animals asso- 

 ciated with water supplies, opens with an account 

 of some experiments made in 1886 on the pipe- 

 fauna of Hamburg, then supplied with unfiltered 

 water from the Elbe. Examples of as many as fifty 

 genera, representing most of the main groups of 

 the animal kingdom, were obtained, and the 

 author gives an interesting account of the life- 

 history of some of the more important, show- 

 ing how when once established they can rapidly 

 spread to the whole of the system, and in some 

 cases — for example, sponges and molluscs — cause 

 grave restrictions to the flow of water in the 

 pipes. 



Under the second heading of " Plants in Water- 

 works " the author deals chiefly with algae and bac- 

 teria. The former class, when present in exces- 

 sive amounts, may cause serious choking of filter- 

 beds, and sometimes give rise to unpleasant tastes 

 and odours, but otherwise are an important factor 

 in efficient filtration. 



Ordinary bacteria, including those which cause 

 water-borne disease, are not dealt with, but a 

 very full account is given of the dreaded creno- 

 thrix or iron bacteria. Several water supplies^ 

 both in this country and abroad, notably Chelten- 

 ham, Liverpool, Berlin, and Rotterdam, have 

 suffered from this pest, and the author describes 

 in some detail the history of these visitations, 

 which have had the effect not only of almost en- 

 tirely choking the pipes, but of imparting to the 

 water a deep red colour and an unpleasant odour. 



In the last section the question of biology in 

 relation to water purification is dealt with. The 

 theory and practice of slow sand filtration, de- 

 pending as thev do on the formation of a biological 

 film on the surface of the sand, are fully described, 

 and the section concludes with a brief summary 

 of the pioneer work of Dr. Houston on storage. 



The pamphlet is profusely illustrated with 

 diagrams and photographs, and is a most useful, 

 interesting, and readable work. 



