[ 3^8 ] 



3. These quantities are formed by multiplying together a number 

 V of the feveral fluxions of x, fo that the fum of their exponents 



p q r t 



iliall be n. Thus \i ab c d be one of thefe quantities /> + 2 §■ + 

 3 r + 4 X = « and p ■\- q -Vr •{■ s = v. 



Note — .By exponent of a fluxion is meant its order. Thus the 

 exponent of ^ or of the fourth fluxion of x is 4. 



p q r s a 



4. To the quantity <7 i^ r (S' - - x is to be annexec^, for a 



coefficient, a fraiftion the numerator of which is n. n — i. ?z — 2 - 

 /^+i, and the denominator /> X ^ — i. - - i x z. q. q — i 



1X3. 2'.r.r — I - - I X 4- 3- 2IX J. j- — i _ _ i 



X 



-I 



ik.k — I - - 1I xfl-'^ff- — I - - I. The law of continuation of 

 which is evident*. 



The DemoTiJiratioTJ, as far as regards the i", a^ and s** laws of the 

 feries, is readily deduced from confidering the manner, in which the 



m 



fucceflive fluxions of x are derived. The demonftralion of the 

 fourth law is fomewhat more difficult, but may be deduced as 

 follows : A quantity a b prefixed to a power of x is evidently de- 



rived by taken the 'fluxion of x^ p + q times, and of a,_q times in 



every 

 * Since writing the above I find that Dr. Waring, at the end of his " Meditationes 

 Analytics," fpeaking of " me^hodus dedu£lionir. & redudtionis," mentions this 

 problem, and givrs the thrc'^ firft laivs, in which indeed there is no diiHcuJty; the 

 fourth, the only one difficult to invcftigate, he does not give, nor does he mentioR 

 any ufe to which the prohlera may be applied. 



