9th and 25th best the 3rd and 4th! This of course is an extreme case: but anything within these limits is possible: e.g. any competitor, from the 3rd best to the 17th best, may, by the mere accidental arrangement of pairs, and by no means as a result of his own skill, carry off the 2nd prize. As a mathematical fact, the chance that the 2nd best Player will get the prize he deserves is only 16/31sts; while the chance that the best 4 shall get their proper prizes is so small, that the odds are 12 to 1 against its happening!

If any one thinks that, after all, we are merely introducing another element of chance into the game, and that no one can fairly object to that, let him try the experiment in a rifle competition. Let him interpose when the man, who has made the 2nd best score, is going to receive his prize, and propose that he shall draw a counter from a bag containing 16 white and 15 black, and only have his prize in case he draw a white one: and let him observe the expression of that rifleman's face.

3. A proof that the present method of scoring in matches is constantly liable to lead to unjust results.

To prove this, let us suppose a `set' to mean `the best of 11 games' and a `match' `the best of 5 sets': i.e. `he, who first wins 6 games, wins a set; he, who first wins 3 sets, wins a match.'

Suppose A and B to play the following 50 games (`A2' means A wins 2 games, and so on) --

B2A5B4|A6|B3A5B2A^*|B^*A2B4A3B|B2A5B34.

Here A wins 28 games to 22, and also wins the match.

But, by simply transposing A^*, B^*, we get

B2A5B4|A6|B3A5B3|A3B4A3|B3A5B3, the last game of the original series not being played.

Here A still wins 27 games to 22: yet he loses the match!

4. A system of rules for conducting Tournaments, which while requiring even less time than the present system, shall secure equitable results.

The method for conducting Tournaments, which I have to propose, involves two departures from the present method. First, I propose to make a `match' last only half a day (the necessary reduction in the number of games I will discuss in section 5): secondly, I propose to give only 3 prizes. The rules for a Tournament of 32 Players would be as follows --

(a) The Tournament begins in the middle of the 1st day, so that there is only one contest that day -- the 32 Players being arranged in 16 pairs.

(b) A list is kept, and against each name is entered, at the end of each contest, the name of any one who has been superior to him -- whether by actually beating him, or by beating some one who has done so (thus, if A beats B, and B beats C, A and B are both `superiors' of C). So soon as any name has 3 `superiors' entered against it, it is struck out of the list.

(c) For the 2nd day (morning) the 16 unbeaten men are paired together, and similarly the 16 with 1 superior (the Losers in these last-named pairs will now have 3 superiors each, and will therefore be struck off the list). In all other contests they are paired in the same way: first pairing the unbeaten, then those with 1 superior, and so on, and avoiding, as far as possible, pairing two Players who have a common superior.

(d) By the middle of the 3rd day the unbeaten are reduced to two, one of whom is certainly `First-prize- man'. These two do not contend in the afternoon contest that day, but have a whole-day match on the 4th day -- the other Players meanwhile continuing the usual half-day matches.