NA TURE 



[February i6, 1905 



the trust-like company of the Loire was formed, that 

 was the prototype of the coal trusts and sj'ndicates 

 of to-day. Events such as these had a far-reacliing 

 influence on the development of the coal-mining 

 industry. 



Special commendation is due to the author for the 

 scrupulous accuracy with which references tb original 

 authorities are given, and for the care with which 

 the proof-sheets have been read. Two trifling mis- 

 prints have, however, escaped detection. Freiberg 

 appears as " Freyburg " (p. 292), and Sir Marc 

 Isambard Brunei as " M. J. Brunei " (p. 291). 



Bennett H. Brough. 



MATHEMATICS OF BILLIARDS. 

 Billiards Mathematically Treated. By G. W. Hem- 

 ming, K.C. Second edition. Pp. 61. (London : 

 Macmillan and Co., Ltd., 1904.) Price 35. 6d. net. 



MORE fortunate, or more careful, than most 

 authors, Mr. Hemming, whose recent death 

 will be regretted by many, did not find it necessary 

 in his second edition to make any material alterations 

 in his original work. He added two appendices, iii. 

 and iv. , with which alone it is necessary to deal in the 

 present notice. 



Appendix iii. discusses the comparative advantages 

 of fine and through strokes, with regard to the margin 

 of error permissible in the respective cases. In the 

 figure opposite p. 47, A is the player's ball, O the object 

 ball, and the stroke is to make A, after striking O, 

 pass within a distance of the point P depending on 

 the nature of the stroke, namely, for a cannon a 

 distance equal to the diameter of a ball, for a losing 

 hazard the necessary distance from the centre of the 

 pocket, which may vary between different tables. 

 The angle AOP is given by the conditions of the 

 problem, and in the notation adopted is t— a. The 

 angle of aim, OAS, is the thing to be determined. It 

 shall be denoted by a, as in appendix ii. of the first 

 edition. In the present appendi.K A; is also used for 

 the same angle. S denotes the position of the centre 

 of the striking ball at impact, SO being the common 

 normal. If ASO = t — e, e and a are connected by the 

 relation sin e/sin o = AO/OS = AO/2 if we denote OS, 

 the diameter of a ball, by 2 ; an4 in the special case 

 considered of AO = PO = 30, or 15 diameters, we might 

 to a very near approximation use a instead" of sin a. 

 Further, the angle OPS is denoted by Pj, and the 

 angle of deviation, tt — ASP, by 5. It is then shown 

 that as the equation connecting S and 9, 



tan {e + S) = p tan 6, 

 where, for reasons given in the former edition, 

 p = 3.5. From this last equation 5 may be obtained in 

 terms of 9 or a. In fact, 



tan 5 = {p-i) sin 9 cos fl/(cos= e + p sin= 9) 

 is easily found. 



The complete method, were it practicable, would be 

 to find an equation in fi or a having two roots, one of 

 which, say 9^, should correspond to the fine, the 

 other, Oj, to the through, stroke, and thence the margin 

 of error might be found for each stroke. This analysis 

 being difficult, a practical solulion is obtained by means 

 NO. 1842, VOL. 71] 



of a diagram in which the ordinate y represents sin A, 

 given by the conditions, and the abscissa x represents 

 sin 9 in an actual stroke in which, for given A, the ball 

 A passes over or verv near to P. A series of values of 

 sin 9 being found corresponding to a series of values 

 of sin A, we draw a freehand curve through them. 

 In general, a line parallel to x for given y cuts this 

 curve in two points, namely. P„, in which 9 has the 

 smaller value (the through stroke), and Q„, in which 

 it has the greater value (the fine stroke). It comes 

 next in order to find for any y the margin of error for 

 P„ and for Qj,. This is done by using the formula of 

 appendix ii., first edition. The linear error on the 

 object ball is (AO being 30) 3080. The consequent 

 linear error at P (PO = 3o) is denoted by E. Then 

 3o5a/E gives the margin of error. A new curve, called 

 the blue curve, is then drawn, having for abscissa 

 .v = sin 9, and for ordinate y = 3o5a/E, in the same way, 

 by a series of trials, as the first curve. The blue curve 

 has two branches. Then the margin of error for any 

 of the points P„ or Q^ of the first curve is that ordinate 

 of the blue curve which has the same abscissa. .As 

 the result of this method it is found that the margin 

 of error is the same for the through as for the fine 

 stroke, when sin a =0320, and sin ^ = 0-132 for the 

 through, and sin 9 = 0-960 for the fine stroke. For 

 smaller values of a the through stroke has the 

 advantage; for larger values of a the fine stroke, 

 until a certain maximum is reached. 



In appendix iv. , /, the coefficient of friction between 

 two balls at impact, formerly taken as zero, is assumed 

 to have the values o-oi or 0-02, and it is found that, 

 instead of p = 2.^, as above assumed, we should have 

 for / = o.oi />' =3.445 + 0.0625 cos 6 

 for / = oo2 /i' =3.391+0-125 cos 6. 

 It will be observed that both these values of p' give 

 very approximately />'=3.5 when ^ = 30°, that is, for 

 the half-ball stroke. 



Before this notice was in type Mr. Hemming was 

 taken from us by death, to the sincere regret of his 

 many friends, including the present writer. 



S. H. BURBURY. 



.4 MORPHOLOGY OF THE ALGJE. 

 Morphologie und Biologic der Algen. By Dr. 

 Friedrich Oltmanns. Vol. i. Special part. Pp- 

 vi + 733; illustrated. (Jena: Gustav Fischer, 1904.) 



THE charming little university town of Freiburg 

 has been the birthplace of important ideas in an 

 obscure department of natural history. De Bary 

 began there his researches into the life-history of the 

 lower fungi, and afterwards continued them at Halle 

 and Strassburg. Owing to his great work and in- 

 spiration we botanists owe a germ-theory of disease — 

 a theory which was in time to bear fruit in practical, 

 medical and surgical form in the mighty hands of 

 Lord Lister. To Freiburg, then, we come again for 

 a morphology of the kindred group of the Algae. 



There is a difficulty in understanding how even an 

 assiduous German professor, living so remote from the 

 sea as Freiburg is, can have obtained the inspiration 

 whicli has guided his research for years past. The 



