Manchester Memoirs, Vol. li. (1907), No. 1<>. 5 



total result of the first throw are exactly the same as half 

 of those which determine the total result of the second. 

 The two throws are connected together. For instance, let 

 us consider the maximum possible difference between the 

 two connected throws and compare it with the maximum 

 possible difference between two independent throws such 

 as we started by making. The maximum possible differ- 

 ence between two independent throws is twelve. A 12 

 may follow a 0. Or a may follow a 12. 



But in the case of two connected throws the maximum 

 possible difference is 6. It may happen by all the red 

 dice showing 4-or-more and all the white ones 3-or-less 

 in the first throw and by all the white ones showing 

 4-or-more when thrown again (that is, by a 12 following 

 a 6) ; or by all the dice showing 3-or-less in the first 

 throw and the white ones all showing 4-or-more when 

 thrown again (that is, by a 6 following a 0). The number 

 of ways in which the maximum difference between the 

 two throws may be attained is given by the number of 

 pairs of figures that follow. The first figure in each pair 

 indicates the result of the first throw in that pair ; the 

 second that of the second. 0-6, 1-7, 2-8, 3-9, 4-10, 5-1 r, 

 6-12, 7-1, 8-2, 9-3, 10-4, 1 1-5, and 12-6. The essential 

 point is that 6 is the maximum possible difference. 



But you see how seldom it is likely to occur. It depends 

 on all the white showing the opposite kind of face upper- 

 most in the second throw to those which they exhibited 

 in the first. The fact that it does not occur often, however, 

 does not concern us now. What concerns us at present is 

 that the maximum possible difference between the result 

 of a pair of throws connected in the way we described above 

 is 6, whilst that between two unconnected throws is 12. 

 The two results in the ' connected ' pairs are, as it were, 

 chained together. We may compare the two results in 



