iOO Application of the Fluxional Ratio, &fc. 



It may here be noticed that the elements p, q, r, s, Uc. in the se- 

 ries fx-\-ph -{-qh^-i-rh^+ he. represent the several functions of .r 

 together with their coefficients, in expanding the binomial {x-\-hy. 



n n— 1 

 Hence fx^rph + qh-" + rh^ -\-sh'-\-hc.=x"-{-nx"-^h+:^- —^^ 



n n — l w— 2 o , • . • r • 



^"-2^2 ^ -. — - — -^a;"~^A^+&c. . substituting x+t ior x, it 



n n — 1 n n — 1 



becomes (x+i)" -}-n{x + iy-'h+Y ~2~^^ "^ ^^^ ^'*'' "^1*^"* 



^-^{x+iy-'h'-{-&Lc.={x+iY+a{x-\-iy-'hi-f3{x-\-iy-^h' + 



r{x+iy-^h^-\-^{x-{-iy-'h'+hc. 



The term (x+iy being expanded forms the equation, 



n n -1 _ . 



{x+iy=x''+nx''-H-\-j^'—^x" 't" + hc. 



Put n— 1 for w, and the equation becomes, 

 n — l n — ln — 2 



Put n — 2 for n, and the equation becomes, 



n — 2 7i~2 n — o 



. Put w — 3 for w, and the equation becomes 



n-3 w — 3 w— 4 . „ . 



hence a;" + aa:""H*+ fSx^'-^i^ =fx+pi + qi^ 



n — l 71 — I n—2 



OCX"- ' +a — - — x^-^ioL— a;" - sjs ^p-j-p'i-fyi^ 



n — 2 w — 2 ?i — 3 



jSaj-'-^-f/S-y— as^-^iiS— -j ^ x^-H^^q + qi-^-q'^i^ 



n — S n — 3n — 4 



y3."-3_{_y-— — x"~'^iy — - — —-- x""- ■H'^ =r + r'tr^r''i^ 



n —4 n — 4 n —5 



1 1 ,i 



n—l 

 therefore /((a; + i)+A)=a;" + ax"-'i+ /3x''-2z3 4-(ax"-'+a-j- 



n-I ?i-2 . , ,, ■ » —2 . . «-2 

 a.'- = e-f-a-^ ^x-H^')h + {rdx'-^ + ^— j-.t^^j -j- /3— j 



^ 2 



-^-a;".-''t = )A2^_&;c.=/a;4-pi+5t+SiCo + (p+/i+/'»')^+ C?- + 



fi+q''i')k^- +(r+r'i.-{'r''i')h' ~^(s+s'i-\-s''i')h' -i-hc, (3,) 



