328 BEGUELIN. 



603. We now consider Beguelin's two memoirs. These as we 

 have stated are contained in the same volume as Euler's memoir 

 noticed in Art. 438. The memoirs are entitled Bur les suites ou 

 sequences dans la lotterie de Genes ; they occupy pages 231 — 280 

 of the volume. 



604. Beguelin's memoirs contain general Algebraical formulae 

 coinciding with Euler's, and also similar formulae for the results on 

 John Bernoulli's conception; thus the latter formulae constitute 

 what is new in the memoirs. 



605. We can easily give a notion of the method which 

 Beguelin uses. Take for example 13 letters a, h, c, ... i,j, h, l, m. 

 Arrange 5 files of such letters side by side, thus 



,m m m 771 m 



Consider first only two such files ; take any letter in the first 

 file and' associate it with any letter in the second file ; we thus 

 get 13^^ such associations, namely aa, ah, ac ... ha, hh, he, ... 



Here we have ah and ha both occurring, and so ac and ca, and 

 the like. But suppose we wish to prevent such repetitions, we can 

 attain our end in this way. Take any letter in the first file and 

 associate it with those letters only in the second file, which are in the 

 same rank or in a lower rank. Thus the a of the first file will be 

 associated with any one of the 13 letters of the second file ; the h of 

 the second file will be associated with any one of the 12 letters 

 in the second file beginning with h. Thus the whole number of 



13 X 14 



such associations will be 13 + 12 + . .. + 1; that is .. ^ . 



' 1.2 



Similarly if we take three files we shall have 13^ associations 



if we allow repetitions ; but if we do not allow repetitions we 



13 X 14 X 15 



shall have — ^ — ^— , Proceeding in this way we find that if 



JL X ^ X o 



there are five files and we do not allow repetitions the number of 

 13x14x15x16x17 



associations is 



1x2x3x4x5 



