March i, 1900] 



NATURE 



411 



"A rule of thumb," he says, "is as good as a scientific 

 law to a man who has played often and well enough to 

 regard the rule of thumb as a necessary law of nature. 

 Amateurs of less experience than this may find it much 

 easier to obey a law the reason for which they have 

 grasped." 



The possible motions of a billiard ball here discussed 

 are five, viz. : (i) Perfect rolling ; that is, rotation about 

 a horizontal line through the point of the ball touching 

 the table for the time being. This is the motion assumed 

 by the ball when struck by a horizontal cue in a vertical 

 plane through the centre at a height 7/10 of the diameter. 

 (2) Sliding without rotation. (3) Pure side ; that is, 

 rotation about a vertical axis through the centre. (4) 

 Curving motion ; that is, rotation about a horizontal axis 

 through the point touching the table, such axis coinciding 

 with the direction of translation. (5) Imperfect rolling ; 

 that is, rolling as in (i) combined with any of the 

 others. 



These probably exhaust all forms of the motion for 

 gentle strokes. A hard struck ball will probably jump 

 many times before it finally subsides into rolling or 

 sliding, just as a cricket ball neither rolls nor slides much 

 till it is nearly spent. 



The most interesting case occurring in practice is that 

 in which the striking ball with perfect rolling impinges on 

 the object ball at rest. The problem of the motion 

 of the two balls after impact involves the deter- 

 mination of the important constants. The constants 

 are : — (i) The coefficient of elasticity, i - ^, between ivory 

 balls, which, on authority accepted by the writer, is given 

 as about 14/15. (2) The friction, /, between the balls, 

 which is determined as follows : — The object ball, if 

 struck obliquely, acquires from the friction with the 

 striking ball a certain rotation about a vertical axis 

 through the centre. This is proportional to/. With this 

 is compounded a rotation about a horizontal axis due to 

 direct impact, so that the resultant rotation is about an 

 axis inclined to the horizon. And if we can guess the 

 direction of that inclined axis, we can determine /. By 

 using an old red ball, on which are irregular markings, 

 as the object ball, Mr. Hemming says the inclination can 

 be guessed with fair accuracy, and this method gives for 

 /"a value between 1/70 and 1/105. 



A third possible constant is that of the impulsive 

 friction between ball and cloth, denoted by /x. This Mr. 

 Hemming retains provisionally in his formulae, but ulti- 

 mately rejects as inappreciable. 



In the case of impact between ball and cushion, the 

 action of the cushion varies so greatly with the speed, as 

 well as with the direction, of the striking ball, that no 

 constant can be determined. 



Probably the most useful part of the book to the 

 practical player is Appendix II., on the margin of error 

 \\\ billiard strokes, from which even the best player may 

 learn something to his advantage. I select the following 

 instances : in playing a winning hazard, the margin of 

 error is least, and the stroke most difficult, when the object 

 ball is half-way to the pocket. It is found also that the 

 margin of error is smaller in a thin losing hazard than 

 in the corresponding through stroke. — But is not the 

 management of the cue more difficult in the latter case } 

 Appendix I. treats of the efifect of nap on a ball played 

 NO. 1583, VOL. 61] 



with side. It raises the question of the nature of rolling 

 friction. According to Prof. O. Reynolds {Phil. Trans. 

 1876), rolling friction may be reduced to sliding friction^ 

 When a body rolls on a plane, expansion or contraction 

 takes place in the substances immediately in contact, 

 which, or the subsequent restitution, causes one to slide 

 over the other. Sliding is thus always being created 

 and destroyed by friction as fast as it is created. Mr.. 

 Hemming makes a different hypothesis — namely, that 

 the cloth may be regarded as a series of stiff parallel 

 ridges, facing the way of the nap. The rolling ball is- 

 instantaneously in contact with two of them, say, at the 

 points P and P', the line PP' subtending at the centre 

 a very small but finite angle, which depends on the 

 nature of the cloth. Through P and P' pass two re- 

 actions, which intersect (he says) in the vertical througl* 

 the centre at the height 7/10 of diameter, and therefore 

 cause the ball to continue perfect rolling. He is not 

 writing a treatise on rolling friction, and does not there- 

 fore give any a priori reason why the two reactions 

 should intersect at the point stated. A posteriori they 

 must do so, for otherwise the ball could not continue 

 perfect rolling as, in fact, it does. It would not be diffi- 

 cult to show, applying what is known as Thomson's 

 theorem, that on Mr. Hemming's hypothesis, as to the 

 nature of the cloth, the ball would pass from the state 

 of rotation round P to a state of rotation round P' with, 

 diminished energy, and so must continue perfect rolling. 



S. H. BURBURY. 



OUR BOOK SHELF. 



A Rudimentary Treatise on Coal and Coal Mining. By 

 the late Sir Warington W. Smyth, M.A., F.R.S. Eighth 

 edition, revised and extended by T. Forster Brown. 

 Pp. vi -1-346. (London: Crosby Lockwood and Son, 

 1900.) 

 No man did more for the advancement of mining edu- 

 cation in this country than the late Sir Warington Smyth. 

 In 185 1, when the Royal School of Mines was founded,, 

 he was appointed lecturer in mining, and he continued 

 to give his annual course of mining lectures until June 

 20, 1 89 1, when, sitting with his students' examination 

 papers before him, he passed away— dying, as he had 

 lived, in harness. In 185 1 he found the art of mining in 

 the trammels of empiricism ; and, thanks to his wide 

 practical experience and his familiarity with continental 

 practice, he was able in his lectures to evolve order out 

 of chaos, and to arrange heterogeneous facts in a com- 

 prehensive system. Moreover, his work underground as 

 mineral inspector and adviser to the Crown enabled him. 

 constantly to keep his lectures abreast of the times. 

 Unfortunately, he never prepared them for the press. 

 But, while directing the higher education in mining, he 

 was not forgetful of the needs of the elementary student^ 

 and was induced in 1866 to write for Weale's excellent 

 series of rudimentary treatises a little book on coal and. 

 coal mining. This was eminently successful, and seven, 

 large editions were called for. No previous work gave 

 so popular and yet so full and accurate a view of the 

 subject. Written in a delightful literary style, it bore 

 internal evidence of not being a mere extract of books, 

 and afforded attractive reading not only for the un- 

 practised, but also for the experienced mining engineer 

 and geologist. . 



Since the publication of the seventh edition great pro- 

 gress in mining has been made, and the value of the 



