143 



GENERAL EXAMPLES 



m 1 n — 1 11 



Given poAver V+x-=. l+»z:v+Cx+&c.+Sx+T»+&c. 



, »— 1 1 "—I n 



1 -\-'^x+cx-^&iC.+sx — l+w«+Car+&c.+S:c+t.r+&c. 



Remainders* * * * *(T-t).r+&c. 



n It 



(T-t).r = r/.c 



71 — I n n+l 



Given power ]+.r=:l+OTJ:+«S£C.+S.r+ T.r +V.r +&c. 



n — 1 n n — 1 n n+l 



l+yX+&c.+sx+tJc=l+m.v+Scc.+St'+{t+rt )x +vx+ &c ._ 



n n + l 



Remainders* *= *' * (T-t+?-/).i-+(V-v).r+&c. 



n+l (1+1 



( V-v )v ■— rvx 



In the former of these Remaindei'S T — t=r/, and hence 

 in the latter Remainder T^t + rt .: T — 1+?'/=0, Therefore 

 the Coefficient of every power of .r whose Index is not greater 

 than n M'ill be destroyed, and the lowest Index in the latter 



o I -- 3 n n+l 



Remainder is «+l, therefore the Law .r, r, ,r, x, &c. x, x, &c. 

 will be the General Law of the several tei-ms of the Root. 



To 



