574 PHILOSOPHICAL TRANSACTIONS. [aNNO 17Q5, 



down the actual multiplication of several of these binomial factors together, 

 which produce compound expressions of the usual well-known forms, in the 

 manner first given by Harriot, from which he infers as follows. 



4. From the above it appears, that the coefBcient of the highest power of x 

 in any equation is 1 ; but the coefficient of any other power of x in the same 

 equation consists of a certain number of members, each of which contains 1, 

 2, 3, &c. of the quantities a, b, c, &c. Thus the coefficient of the 3d term 

 of any equation, is made up of members, each of which contains 2 of the quan- 

 tities only, as, a6 -j- ac + be, the coefficient of the 3d term in the cubic equa- 

 tion. And indeed, not only from inspection, but also from considering the 

 manner in which the equations are generated, it is evident, that each member of 

 any coefficient has as many of the quantities in it, as there are terms in the 

 equation preceding the term to which it belongs. Thus, abc -\- abd -f- acd -f- 

 bcd is the coefficient of the 4th term in the biquadratic, each of the members 

 has 3 quantities in it, and 3 terms precede that to which they belong. 



5. When any equation is multiplied in order to produce the equation next 

 above it, it is evident that the multiplication by x produces a part in the equation 

 to be obtained, which has the same coefficients as the equation multiplied. Thus, 

 multiplying the cubic equation by x we obtain that part of the biquadratic which 

 has the same coefficients as the cubic: the only effect of this multiplication being 

 the increase of the exponents of x by I. 6. But when the same equation is 

 multiplied by the quantity adjoined to x by the sign -f? each term of the product, 

 in order to rank, under the same power of x, must be drawn one term back. 

 Thus when the first term of the cubic is multiplied by d, the product must be 

 placed in the 2d term of the biquadratic. When the 2d term of the cubic is 

 multiplied by d, the product must be placed in the 3d term of the biquadratic; 

 and so of others. 



7. As the equation last produced is the product of all the coinpound quantities 

 X •{■ a, X + b, X -\- c, &c. into each other, and as it was proved in the 2d 

 article that each of the quantities a, b, c, &c. must be found the same number 

 of times in this product, if we can compute the number of times any one of 

 those quantities enters into the coefficient of any term of the last equation, we 

 shall then know how often each of the other enters into the same coefficient: 

 and this may be done with ease, if of the quantities a, b, c, &c. we fix on that 

 used in the last multiplication. For the last equation, and indeed any other, 

 may be considered as made up of 2 parts; the first part being the equation im- 

 mediately before the last multiplied by x, according to the 3th article, and the 

 other being the same equation multiplied by the quantity adjoined to x by the 

 sign -|-, last used in the multiplication, according to the 6th article. This last 

 used quantity therefore, never enters into the members of the coefficient of the 

 first of these 2 parts, but it enters into all the members of the coefficients of the 



