E

H

| D

M

K

B

the line of direction makes with a line drawn perpendicular to the face of the ivory block. This angle is called the angle of incidence. The line along which the ball travels when returning from the cushion is called the line of reflexion, and the angle this line makes with the same perpendicular is the angle of reflexion. In perfectly elastic and hard bodies the angle of reflexion would be equal to the angle of incidence. It is so in the case of the cosmical ether which conveys the motions of light and heat; but no material substance is perfectly elastic, and therefore in the case of ivory we must make the same kind of allowance that we had to do when speaking (Section 5) of the angle of separation of impinging balls. In the case of an immoveable block, as in that of a moveable ball, the motion of the arrested ball is resolved into two, one in the direction of the resistance, the other perpendicular to it. In the case of collision between balls, we have seen that the first of these was transferred wholly to the red ball. In the case of an immoveable block, a portion of this motion is restored to white in the opposite direction; the remainder is lost, or rather converted into heat, owing to the imperfect elasticity of ivory. The portion of the resolved motion so restored, combines with the motion perpendicular to the resistance direction, to describe the actual path of the rebounding ball. To ascertain the path we must draw the figure in the margin. In this, A B represents the face of the block of ivory, which is substituted for the usual cushion. EC is the line of direction, and C D a line drawn through C perpendicular to the cushion. A ball advancing from E, and striking the ivory block at C, has its motion resolved into one represented in strength and direction by G C, and another which is represented by C F made equal to E G. This latter part of the resolved motion is not affected by any imperfection in the ivory, but the other part, though restored in an opposite direction to G C, that is in the direction of C G, loses from one tenth to one fifteenth of its force. Its strength will therefore have to be represented by a line, in this proportion shorter than G C, measured from C, in the direction C G. This line is C H, and the motion represented by it combines with the residue of the original motion of white represented by C F, to form the line of reflexion CK, representing the strength and the direction taken by the rebounding ball. It is evident that the angle D C K, or the angle of reflexion, is greater than the angle D C E or the angle of incidence, and this is always the case with hard elastic substances, such as ivory, glass or steel. It should here be noted that a surface of ivory such as we have here supposed, has no effect on any motion of rotation possessed by the incident ball before collision, so that it remains round the same axis as before. Thus, if the ball had a following motion of rotation before striking the block the rotation would after reflexion no longer be in the same direction as the altered course of the ball, but transverse to it, and as the friction of the table began to act on the ball after reflexion its course would be altered into the form of the curve C L, while an original back rotation or screw would produce the curved path C M. Any side or rotation round a perpendicular axis would remain unaltered, and would only cause it to continue to spin until the friction of the table brought it to rest.

The reader must remember that all these conclusions depend on the supposed substitution of an ivory block for the usual cushion of a Billiard table. Such cases only are treated of in mechanics. But the effects we have described are seldom seen on a Billiard table, except when a ball touching the cushion is struck by the white at the opposite extremity of the diameter which so touches. In this case alone does the white act as it had struck our supposed block of ivory. When struck with medium strength, high and with a line of direction inclined from 20 to 30 deg. to this diameter, the curve described by white is so marked as greatly to surprise those whose knowledge and experience has been derived wholly from the Billiard table.

SECTION 26.-FAILURE OF THE THEORY.

True as these conclusions undoubtedly are, under the stated conditions, a very short experience at the Billiard table is enough to convince any one who attempts to explain the phenomena by the theory, that some circumstances must have been omitted in the calculations, as these by no means account for the effects observed. The angle of reflexion of a Billiard ball after striking the cushions actually in use, instead of being greater, as it ought to be by the theory, is generally very much less than the angle of incidence, and the rotation of the ball round a horizontal axis does not produce the curved paths shown in the figure. On the other hand, rotation round a perpendicular axis has a very marked effect on the angle of reflexion, causing the

actual line of reflexion to vary often as much as 30 deg. to one or other side of the line, which would be the result of the construction according to Fig. 4. We must therefore examine into the causes of the difference between theory and practice, and endeavour, by supplying the neglected conditions, to account for the actual motions we observe in a ball rebounding from the cushion of a Billiard table.

SECTION 27.-CAUSES OF THE FAILURE.

There are three principal causes of difference between the conduct of a hard elastic material on the surface against which a ball strikes, and a material so soft and yielding as india-rubber, covered with fine cloth. The first of these is the time occupied before the force of restitution, which causes the return of the ball, comes into play. Elasticity is a mechanical effect produced by a change of shape in one or both of the bodies coming into contact-compressing the substance into a state of constraint, from which it seeks to relieve itself by the reaction of internal forces, which act to a greater or less degree, and with greater or less rapidity according to the nature of the substances. In the cases investigated by mathematicians, the substances were very hard, such as ivory, glass and steel, and the reaction of the internal forces bore a large proportion to the amount of compression, and the time of contact was so short as to produce no appreciable effect on the rotation of the impinging balls. In Thompson and Tait's "Elements of Natural Philosophy," Section 259, they say, "When two bodies in relative motion come into contact, pressure begins to act between them to prevent any parts of them from jointly occupying the same space. This force commences from nothing at the first point of collision, and gradually increases per unit of area on a gradually increasing surface of contact. If, as is always the case in nature, each body possesses some degree of elasticity, and if they are not kept together after the impact by cohesion or by some artificial appliance, the mutual pressure between them will reach a maximum, will begin to diminish, and, in the end, will come to nothing, by gradually diminishing in amount per unit of arc, in a gradually diminishing surface of contact. The whole process would occupy not greatly more or less than an hour if the bodies were of such dimensions as the earth, and such degrees of rigidity as copper, steel or glass. It is finished, probably, within a thousandth of a second if they are globes of any of these substances not exceeding a yard in diameter."

In the case of Billiard balls, the time would be very much less than even this ;-the balls have parted company long before any observable effect has taken place on any rotatory motion they may have possessed. The next cause of difference is the friction exercised by the cloth which covers the cushion in arresting sliding motion, and the third is the form and amount of the depression made in the yielding rubber, by an ivory ball striking against it with some force. We shall now endeavour to find the effects of these causes on a ball struck against the cushion with any of the strokes distinguished in Section 7, or having any kind of rotatory motion in addition to the direct motion. For convenience sake, we shall consider separately the two cases of a ball advancing perpendicularly to the cushion and obliquely.

SECTION 28.-STROKES PERPENDICULAR TO THE CUSHION.

A ball, having a direct motion only, striking the cushion at right angles, returns along the same line by which it advanced. There is in this case, nothing in the peculiarities of the cushion to cause it to diverge from this line, either to right or left. On leaving the cushion the friction of the table has the same effect upon it, as we described it to have on a ball struck true centre in the middle of the table (Section 10), that is it causes it to take a rolling motion, towards the player. If the ball on striking the cushion had a motion of rotation round a horizontal axis, that is, if it have either follow or screw, this motion is arrested in the interval, short as it is, during which the ball has been in contact with the cushion. For the process described by Thompson and Tait as so rapid in the case of hard or rigid bodies, takes an appreciable time when the substances which come into contact are of ivory and india-rubber, and during this time the rotation in the ball will be of the nature of a sliding motion at two points, one where it is in contact with the cloth of the cushion, and the other with the cloth which covers the bed of the table. Thus the great opposition to sliding motion possessed by the cloth, acts under circumstances of great power, and we know from experience that the time of contact is long enough to enable it to arrest completely any rotation it is in our power to give with the cue. If the rotation is round a perpendicular axis, the friction of the cloth which covers the cushion, has still a great effect upon it, but of a very different kind. The motion of the ball is stopped only at the one point where it touches the cushion. The angular velocity is sufficient to overcome the much smaller amount of friction of the cloth covering the table, and the rotatory motion is converted into a rolling one along the face of the cushion. A rolling motion we have seen to be a combination of direct and rotatory motion, and the direct part of this case of rolling combines with the restored motion to form a path for the rebounding ball inclined to the direction which it would otherwise take. The amount of this inclination depending on the proportion between the two kinds of motion in the ball at the moment of striking, is due to the manner of the stroke. This is what is called side, an effect of great importance to the billiard player, since the use of leather pointed cues roughened with chalk, have enabled him to give a rotatory motion to the white ball. By practice in the use of side, we are able to vary the angle of reflexion to a very great extent without altering the angle at which red is struck. It will be seen that this effect is the result of the imperfection of the material of which the cushions are formed. In

ΙΟΙ

the same way the effects of screw and following strokes are obtainable only owing to the friction of the cloth, which may be considered an imperfection of the table. It may be here remarked that the greater part of the rotatory motion is exhausted by its conversion into direct motion, although a small portion remains, round the same axis as before and in the same direction.

ANSWERS TO CORRESPONDENTS.

We particularly request all our correspondents to address their letters to the Editor, 67 Barbican, E.C., and not to any one who is, or who is supposed to be, connected with the Papers.

NOTICE. The names of those Subscribers who have not paid for Vol. VI. by the 24th September, will be erased from our List.

THE RETURN of the English players from Vienna was the event of the past month. The result of the great battle is now known all the world over-was known, indeed, on the morning of the 30th of August, from the farthest corner of India to the United States of America-and the victory of Steinitz has been discussed among Chess-players of every clime, so that it is unnecessary to recapitulate it here. The tidings, announcing the defeat of Blackburne by Rosenthal, undoubtedly caused some disappointment in this country; the remarkable success of the Englishman up to the last "round" justifying the hopes of his countrymen in his ultimate victory over all his competitors. The loss of this match is ascribed to temporary indisposition, and the explanation may be accepted without any unfairness to M. Rosenthal, who, although unsuccessful in his matches with the other prominent champions, is a player of great force and ingenuity. To his prowess, at all events, Mr. Steinitz is indebted for the opportunity which gave him the victory; and if, as is reported, he has been munificent in his offerings to the good genius of the occasion, we see nothing to condemn in the gratitude which inspired him. It is not in any grudging spirit to Mr. Steinitz that we venture to express our sympathy with Mr. Blackburne, in his capacity of Victim to the freaks of fortune, for the feeling has a patriotic not a personal origin. Say what we will about the English training, and even proclivities of Mr. Steinitz, we cannot altogether ignore his nationality; and when his compatriots declare that he is but a borrowed sprig in our wreath, candour compels us to admit the truth of the assertion. To do his countrymen, justice, they have not been slow in claiming him, nor in conferring upon him all the honours his conquests deserve. Immediately after the termination of the Tourney a dinner was given in his honour by the members of the Vienna Chess Club, at which the orations in his praise were enthusiastically received. It strikes us as somewhat remarkable that none of the foreign competitors were present on this occasion; but it is satisfactory to note that they were not forgotten-the toast of their health having been received with every mark of cordiality.

The Vienna Shachzeitung publishes, this month, the annexed problem, composed by R. Willmers, and dedicated to the four prize bearers. It is in our opinion a remarkably fine study, and we commend it to our problem solvers. It would be a striking injustice to dismiss the theme of the Vienna Tourney without offering tribute to the management which so enhanced its success. Herr Kolisch has displayed throughout the Tourney the very highest administrative powers, and has set an example of adherence to accepted rules and regulations which may be studied elsewhere with profit to competitors and the public. His kindness and courtesy to the English players they have warmly testified to since their return, and we observe that M. Rosenthal has addressed a letter to La Stratégie bearing witness to the same effect. As the complete score only appeared in our 3rd edition of last month, we reproduce it on page 105.

BLACK.

WXITE.

A short match for nominal stakes is to be commenced at once between Messrs. Bird and Blackburne. The player who first wins seven games to be the victor.

For the following paragraph we are indebted to Mr. Paul Schultz, the honorary secretary of the association referred to :

On Tuesday evening, the 16th ult., the Chess Club connected with the Schiller-Anstalt, Oxford Road, Manchester, inaugurated the coming winter season with a Soirée, combined with an exhibition of objects bearing relation to Chess and various other games. Almost every variety of Chessmen and Boards, from the approved Staunton pattern adopted in this country, to the primitive appliances used in the remotest parts of the Globe, were represented in the show-a magnificent set of carved ivory Chessmen of Chinese workmanship attracting particular attention. An excellent supper, provided by the steward of the establishment, contributed largely to the enjoyment of the evening, and play was kept up with great spirit till a late hour. The Club comprises about forty members, recruited exclusively from subscribers to the Schiller Institution.

The idea of an exhibition of Chess pieces and boards is a very good one, and we should like to see it taken up by the City of London Club, under whose auspices a great success might be anticipated.

The Bow and Bromley Chess Club, although but twelve months old, promises to become an association of some importance in Metropolitan Chess circles. The Club possesses a goodly number of active members, and in set matches has already defeated the City Bank, Eclectic, St. Marks and (in one encounter) the old established Bermondsey Clubs. In a second match with the latter, Bow was defeated, but only by nine games to seven, two being drawn, a proportion that argues little inferiority on the part of the losers. The Hononary Secretary is Mr. J. Foster Mainland, and the members meet every Wednesday and Saturday evening, at the Bow and Bromley Institute, Bow.

The following paragraph is from a correspondent :

The Series of Matches for the Trophy offered by the Working Men's Club and Institute Union terminated on Saturday, 6th September, in favour of the Bedford Institute Working Men's Club, Shoreditch. The Competitors were-Mr. S. J. Stevens, President of the Bedford Chess Club, and Mr. H. Shaw, Representative of the Croydon Club. The Croydon Competitors had vanquished the representative of the Bermondsey Chess Club, while the Bedford representative had defeated the Delegates of St. John's Club, the North Woolwich Club, St. Andrew's Club, and now, having defeated the Croydon Club, has succeeded in securing the Trophy, without the loss of a single game.

We shall endeavour to find space for the deciding game in our next number.

The Recreationist (edited and published by J. White, Leeds) advances in public favour. It is well printed upon good paper, and the games of Chess and Draughts are ably treated by contributors and editors. The Huddersfield College Magazine, besides an ingenious-two move problem by Mr. Greenwood-whose name we have missed from the Chess Worid for some time past-contains an interesting melange of problems, games, and reminiscences of the editor, entitled Chess in the Holidays. It is a capital article for a wintry evening, and likely enough to banish painful reflections upon the price of coal, and set pleasant faces in the fire for every Chess player.

The Newcastle Weekly Chronicle opened a Chess corner on the 20th ult. With such players as Mitcheson and Charleton, Tyneside Chess should be well supported.

Messrs. Mead and Co., 73 Cheapside, have issued a double pocket Chess-board.

In dimensions

and general appearance it is an improvement upon the old single one; but it is most certainly spoiled by the figures upon the squares, and the arched aperture for the moveable Chess-pieces, which, combined, have a most confusing effect upon the eyes. In colouring the squares too, black might be substituted for pink with advantage. With these defects remedied, as they may be without difficulty, the double pocket Chess-board is certain to become popular.

Sissa de Schaakspeler comes freighted with more than the usual number of games and problems. The latter are all excellent, and the enterprise which placed so promptly before the public twelve games played in the Vienna tourney, with copious notes appended, cannot be too highly praised. To be sure, our admiration of the publisher's enterprise suffers some modification from the circumstance that the games and notes have been copied from this Journal, without acknowledgment. Perhaps "enterprise" is not the proper term in this case.

The Dubuque Journal contains, this month, a portrait and biography of Mr. J. N. Babson, a young problem composer of great merit and striking originality. The other contents are varied and interesting

as usual.

The New York Clipper reprinted, from this Journal, the letters of Mr. Miles and Mr. Menzies on the classification of problems. Appended is a courteous and complimentary request from the accomplished Editor, Miron J. Hazeltine, for the writers' autographs.

A correspondent in Adelaide forwards the Chessikin in the margin. It was played some years ago, and before the winner, Mr. Tyrrell, had seen a book opening. We think it worthy of a place in our collection of "brevities."

WHITE.

Mr. S. Tyrrell.
1. P to K 4
2. B to B 4
3. Q to B 3
4. Kt to KR 3
5. Kt to Kt 5
6. B takes P' mate

BLACK. Mr. C. 1. P to Q Kt 3 2. B to Kt 2 3. Kt to K B3 4. B takes P

5. B takes Q

The first meeting of the Eastern Counties Chess Association will be held on the 5th November, at the Town Hall, Ipswich. A tournament will be held, open to the residents of Norfolk, Suffolk, Essex and Cambridge, and entries should be sent to the Secretary before the 22nd October.

The annual meeting of the Bristol and Clifton Chess Association was held on the 24th ult. the Rev. A. C. Rowley, the President, in the chair. From the report it appears that the Association now numbers 98, a number we think exceeding any other purely Chess Club in England, except the City of London. The Club has, in the past year, beaten Bath twice, and the meeting of the Counties Association was a great success. The finances are in a satisfactory position. It was suggested that the Counties Chess Association should again be invited to Bristol, but some members appeared to think it would be better to invite the British Chess Association in 1875. Thereupon Mr. Burt expressed his opinion of the want of courtesy of the Association, in not informing them what prizes they were playing for, and for so abruptly terminating the match. The meeting separated, after having elected their officers for the ensuing year, without coming to any decision on