472 ON THE SUMMATION OF SERIES, 
F' (x), &¢. &c.; we shall have, putting h+1=h’ 
F(z), F,(«), &e., &e. To effect this, let us take the successive 
derived functions, of equation (4), with respect to h. 
wf! (wh) =F (x) +2h.F (x) 4+3h7.F (x) +4h*. F(x) + &e. &e. 
f’(a-+-h)=2F,(2) +2.3h.F\(x)+3.4.07F,(x)+&e. &e. 
f"(a+h)=2.3.F (7) +2.3.4.2.F,(7)+&c. &e. 
&e., &e., &e. 
These derived functions are true, whatever value be given 
toh. Take h=o, then we have 
F(a)'= f(z), F(2) = a .E,(«) = = &e., &e. 
Substitute these values in equation (4), and we shall have 
h I h II h III 
f(x-+-h) = f(x) as i f(x) + 5° f(x) +753° f(z) + &e., 
Bei, GC. scccsccevccccovesncesscccvescecvece Faesecesne Geeceese seceee(O) 
which is Taylor’s theorem. 
