50 



Major P. A. MacMaJwn 



[Feb. 14, 



Fig. 9. 



eventually been completely solved, and in order to lead you up 

 gradually to an understanding of the method that has jDroved success- 

 ful, I ask you to look at the Latin square of order 5 that you see in 

 the diagram (Fig. 9). The first row of letters can be written in any 

 order, but not so the second row, for each column when the second 

 row is written must contain two different letters. We must, there- 

 fore, be able to solve the comparatively 

 simple question of the number of pos- 

 sible arrangements of the first two rows. 

 For a given order of the letters in the 

 first row, in how many ways can we 

 write the letters in the second row so 

 that each column contains a pair of 

 different letters? This is a famous 

 question, of which the solution is well 

 known ; it is known to mathematicians 

 as the " probleme des rencontres." It 

 may be stated in a variety of ways ; one 

 of the most interesting is as follows : 

 A person writes a number of letters and 

 addresses the corresponding envelopes ; 

 if he now puts the letters at random into the envelopes, what is the 

 probability that not a single letter is in the right envelope? 



Passing on to the problem of determining the number of ways of 

 arranging the first three rows so that each column contains three 

 different letters, it may be stated that up to 1898 no solution of it 

 had been given ; while it is obvious tliat as the number of the rows 

 is increased the resulting problems will be of enhanced difficulty. A 

 particular case of the three-row problem had, however, been con- 

 sidered under the title "probleme des menages" and a solution 

 obtained. It may be stated as follows : — 



A given number of married ladies take their seats at a round table 

 in given positions ; in how many ways can their husbands be seated 

 so that each is between two ladies, but not next to his own wife ? 

 For order 5, that is five ladies, the question comes to this ; Write 

 down 5 letters and underneath them the same letters si lifted one 

 place to the left; in how many ways can the third row be written so 

 that each column contains three different letters ? This particular 

 case of the three-row problem for any order presents no real difficulty. 

 The results are that in the cases of 3, 4, 6, 6 . . . married couples 

 there are 1, 2, 13, 80, etc. ways. 



Since the year 1890, the problem of the Latin square has been 

 completely isolved by an entirely new method, which has also proved 

 successful in solving similar questions of a far more recondite 

 character, and I am here this evening to attempt to give you some 

 notion of the method and some account of the series of problems to 

 which that method has been found to be applicable. 



There is, us viewed mathematically, a fundamental difference 



