[ 329 J 



every different order, with the exception that each a muft be taken 

 before the b^ which is derived from it. Confequently the coeffi- 



cient oi a b muft be the number of thefc different orders. 

 This coefficient may therefore be deduced either from the 

 dodrine of permutations or from that of probabilities. The 

 fxjrmer method is certainly the moft natural, and at firfl 

 fight may appear fhorter : but the latter is more readily ap- 

 plicable to general expreffions. And from it the coefficient of 

 P ? 

 a b \% deduced by finding the probability of taking a, a, a, - (p) 



a, a, a - (p) b, b, b - (q terms) in the order in which 



"they are written. The marks underneath fliewing the a' from 

 which the correfponding b' are derived. The inverfe of the 



fradion expreffing this probability is the number of different 



P 1 

 orders, and confequently the coefficient oi a b . The prob. that 



an a, from which a ^ is not derived is taken firft is — , that ano- 



ft 



ther dof the fame defcription is taken next is " , &c. fo that the 



n — I 



probability that all the a' of that defcription are taken previoufly to 



any of the as^ from whence the b' sue derived is — x ■ ^ ■^^^^^ v 

 ■^ n n—i ^ n--% ^ 

 I 

 -r^:=. That an a is taken next is certainty or 



Vol. VII. T t ^„ 



