VOL. LXXXV.] PHILOSOPHICAL TRANSACTIONS. 575 



last of them. But that part into which it does not enter has the same members 

 as the coefficients of the equation immediately before the last, by the 5th article; 

 and when the members of the first pan are multiplied by the last used quantity, 

 the product becomes the 2d part of the whole coefficient above-mentioned. Thus 



the first part of the cubic equation, by the 5th article is, *'^ T / !- ■^^ + abx, and 



as these coefficients are the same as the coefficients in the quadratic equation, 

 being multipied by c, and arranged according to the 6th article, we have the co- 

 efficients of the 2d part of the cubic, viz. c -^ ac , , tt ,. • •, ^ 

 *^ _L A + "'"^- Hence it IS evident, 



that there are as many members in any coefficient, which have the last used 

 quantity in them, as there are members in the coefficient preceding, which have 

 not the same quantity; and as it has been proved that each of the quantities a, 

 b, c, &c. enters the same number of times into the coefficient of the same term, 

 what has here been proved of the last used, is applicable to each. 



8. From the last article the number of members in the several coefficients of 

 any equation may be determined. For if we put s = the number of times each 

 quantity is found in a coefficient, n = the number of quantities a, b, c, &c. and 

 p = the number of quantities in each member; then as a is found s times in this 

 coefficient, b is found s times in this coefficient, &c. the number of quantities 

 in this coefficient, with their repetitions, will he s X n, and as p expresses the 

 number of quantities requisite for each member, the number of members in the 



coefficient will be -. 

 P 

 g. Using the same notation, we can, by the last 2 articles, calculate the 



number of members in the next coefficient. For as — expresses the number of 



members in the above-mentioned coefficient, and s the number of times each 



quantity is found in it, s = the number of times each is not found in it. 



Bv the 6th artic'.«», therefore, a will be found s times, b will be found — ~ 



J P p 



s times, &c. in the next coefficient, and (-- — s) X n = ^ ~ ^^" =z the num- 



P p 



ber of quantities, with their repetitions, in it. But as the number of quantities 



in each member of a coefficient is 1 less than the number in each member of the 



coefficient next following, each member of the coefficient whose number of 



members we are now calculating, will have in it p -|- I number of quantities. 



Consequently ■ ''" 7 T". n = - X - 7 f = the number of members of the coeffi- 

 ^ ■^ P n ip + i) p p + I 



cient next after that whose number of members is — , as in the last article. 



P 



10. The binomial theorem, as far as it relates to the raising of integral powers, 

 easily follows from the foregoing articles. For if all the quantities a, b, c, &c. 



