July 1 1, 1907] 



NA TURE 



245 



of the mollusca inhabiting the mountain country, and 

 the molluscan fauna of the great delta of the Ganges 

 and Brahmaputra had its origin in the Himalayan 

 slopes, although they have occupied their present 

 quarters for sufficiently long a period to become speci- 

 fically distinct. (BV)=. 



WATER AND THE PUBLIC HEALTH. 



(i) The Value of Pure Water. By George C. 

 Whipple. Pp. viii + 84. (New York: John Wiley 

 and Sons; London: Chapman and Hall, Ltd., 

 1907.) Price 4^. 6d. net. 



{2) Tlie Bacteriological Examination of Water .Sup- 

 plies. By Dr. William G. Savage. Pp. .xvi + 2g7. 

 (London : H. K. Lewis, 1906.) Price 6s. 6d. net, 



(1) T^HIS little book is planned on novel lines and 

 J- deserves recognition. An extract will best 



ser\-e to define its scope. 



" Given two water sources equally available to a 

 city for purposes of supply, both safe to drink, but 

 one high coloured and soft, the other colourless and 

 hard — which is the better selection ? A water-works 

 plant is to be appraised : structurally the system is a 

 good one, but the quality of the water is unsatisfac- 

 tory because of its excessive colour or turbidity — how 

 much should be deducted from the value of 

 the works because of the bad quality of the 

 water? The water-works owned by a private 

 company are to be purchased by the city ; 

 the city has a high typhoid fever death-rate, due 

 unquestionably to the water supply — how much less 

 should the city pay because of that fact? A city is 

 using turbid river water — how much can it afford to 

 pay "to filter it? A city is using a water so heavily 

 laden with Anabaena that it is nauseous to drink — 

 how much can the city afford to pay to procure a new 

 supply? " 



An attempt is made from available data to 

 establish formulae which may be employed to cal- 

 culate the allowable depreciation due to sanitary 

 quality, physical characteristics (colour, odour, &c.), 

 hardness, &c., of a water supply. 



The following example is a calculation in the 

 case of typhoid fever. The financial loss to the com- 

 munity for each death from typhoid fever is assumed 

 from various data to be equal to 10,000 dollars. A 

 proportion of the death-rate is due to the disease 

 transmitted by means (shell-fish, flies, &c.) other than 

 water. Assuming that all typhoid in excess of N is 

 due to the water supply, that the daily consumption 

 of water is 100 gallons per capita, and that T is the 

 total typhoid death-rate per 100,000, then (T-N) 

 10,000 = loss to the community in dollars for 

 365x100x100,000 gallons of water, or 



D = <I^?"^5?=275(T-N), 

 365 

 where D stands for the loss in dollars per 

 million gallons of water used. The author is 

 quite alive to the fact that local and other 

 conditions must modify his conclusions, and recog- 

 nises that more data are required before finality is 

 reached in the elaboration of the formula. The 

 book is suggestive and stimulating reading, the 

 NO. 19(37, VOL. 76] 



various tables add to its value, and we heartily com- 

 mend it to the sanitarian and water engineer. 



(2) This book by Dr. Savage, who has made many 

 important contributions to the subject on which he 

 writes, gives an admirable survey of the present posi- 

 tion of the bacteriology of water supplies. Succes- 

 sive chapters deal with the bacterial content of waters 

 and the influences affecting it, excreta, sewage, and 

 soil in relation to the bacteriological examination, 

 the characters of the intestinal bacteria, bacterial 

 evidences of pollution, and full details of the methods 

 emploj'ed in the bacteriological examination of water. 

 The chapter on the interpretation of results is par- 

 ticularly to be recommended. A full bibliography is 

 appended. The medical officer of health, the analyst, 

 and the bacteriologist will find this book a trust- 

 worthy and useful guide. R. T. Hewlett. 



THREE M.iTHEMATICAL TRACTS. 



Quadratic Forms and their Classification by Means 

 of Invariant Factors. By Prof. T. J. I 'A. Brom- 

 wich, F.R.S. Pp. viii+ioo. (Cambridge: The 

 University Press, 1906.) Price 3s. 6d. net. 



The .ixioms of Projective Geometry. By Dr. A. N. 

 Whitehead, F.R.S. Pp. viii + 64. (Cambridge: 

 The University Press, 1906.) Price 2S. 6d. 



The .Axioms of Descriptive Geometry. By Dr. A. N. 

 Whitehead, F.R.S. Pp. viii + 74. (Cambridge: 

 The LTniversity Press, 1907.) Price 2S. 6d. 



THESE are Nos. 3, 4, and 5 of the Cambridge 

 Tracts in Mathematics and Mathematical Physics, 

 which are intended to help students by providing 

 them with brief and readable introductions to mathe- 

 matical theories which are important in themselves, 

 and yet for various reasons do not appear in the 

 ordinary text-books. U they serve their purpose they 

 will induce their readers to follow up the paths they 

 indicate, and try to explore still further the mazy 

 garden of the mathematical muse. 



The present state of the theory of quadratic forms 

 illustrates very well how much interest there may be 

 in the particular cases of a problem which, in its 

 so-called " general form," has a trite and familiar 

 solution. To put the matter into a geometrical 

 shape ; when there are four homogeneous variables, 

 let S = o, T = o be the equations of two quadratic 

 surfaces; then in general the family S + \T = o will 

 have a common self-conjugate tetrahedron, and by 

 taking this as a tetrahedron of reference, S and T 

 assume a well-known standard form. But there are 

 thirteen other cases to consider, for each of which 

 there is a distinct reduced form of S + xT; for 

 instance, if S and T intersect in a cuspidal quartic, 

 the reduced form is 



2(A + a).ij + 2yz + />{>^ + <J)=- + 1-( A + 'i')'-- 

 If we consider the small oscillations of a dynamical 

 svstem with four degrees of freedom, we are con- 

 fronted by precisely the same analytical problem of 

 reduction; the algebraical classification is the same, 

 but certain cases are ruled out by the condition that 



