An Essay On Algebraic Development Containing the Principal Expansions in Commo

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An Essay On Algebraic Development Containing the Principal Expansions in Commo
Thomas Jarrett
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N, (121) and (14. ;}).
59 172. CoK. 3. D^, y{u) = {D, +B^+D, Dy)u, (iGs) = {D, ^Dy+D, Dyy. U, (121) and (143); = |(i+A)(i+A, )-ii"w.
n + \ \n 173. Theorem. D^^irt)^^, ^-^)-' . -^. El:;^\u).
For, D, y{ii) = E, _y{ii)-u, (168); = {E, . , -\)n.
■■ D::, ^{u) = iE. , y-l)„ u = (£, ^-1)"?/, (121) and (143); n+\ [ n =s, „(-ir-'-^-^. ;T 00- m—\ \n 174. Theorem. £;; (w) = s1t^ •i>:;;(^0- For, E, _y{ii) = u + D, ^y{u), (168) ; -(1+Z), s/)'*; = (1+2), ^)". 7/, (121) and (143); L ?» — 1 60 175. Theorem.
r—
...s 6—1 m, For, E;"(ic) = Sr^'^-D;-'(u), (145) and (13).
\m Ej\E;'{u) = SrT^-Dr'-E;'{u), (168), (134), (135) & (141); _ i m \n = S, -r^. D;-KS, -^. D;-'(7c), (145) and (13); r-1 5-1 \m r-l \n = S, . -^;^. S. T^^. A. '-'. I>;-'(w), (136) and (137); r-l \s-l r [^ • \l — Or Oj ri r-s 4—1 r—s s—1 . D;-\D;-'(7i), (6) and (18).
176. Theorem.
Z), -' (uv)=u. D, ''u-D, r' j2>, (?^). A-'^.^S For, DXii-D^~^v) = uv+D(u). D, -\E, (v), (159) and (142).
u . Dr'v=D, -'(uv)+D, -' \D, (u).


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