lio - Mr . Woodhouse on the Independence of the 
By Art. V. <p [x -|- o) — -f- Po -|- 4- — S-L__ See. 
I o 2 ' I • 2 « j 
Let d be the note of the operation to be performed on <px , in order 
to deduce P, then P = D(px, (J = dP= DDtpx = D a <p x &c. 
Hence, <p [x -j- o) = q> x + d <p x o -f- D -- 1- °- -f- &c. 
or, representing ii by ? ^ 
+ o) 
= pa: -f D • 0 + ?* <p X . O a -|- D 3 <p X . Q* + & c * 
To resume the demonstration : 
x ? i 2 + (« + (3) 2 -j- b x _ 
i -f- « x * x -j- @x i -f (a, + (3) x a & x* i b x + c 
now — - — - = i — a,x 4- cd x 1 a n x n . . » 
I -j- a X 1 
and — b__— = i — » (3 x 4- /3 1 x * . . . . . =f= (3 n x n 
i -1 - pi 8 
term affected with x n in developement of + 7TJ7 
is =±= <j -of -j- }. 
Again- = ( a + 6x ) (1 + bx + cx')~' 
now (i -j- bxf 1 = i~~ 1 -j- r> i 1 .'feor + 9* 1 • 6 2 uf -f? 3 l 1 .6 , o?*+&c. 
for 6 put 6 ^ <r, and for -{- cx^ m i or b m -|- d b m . cx + D-6". c z x* &c. 
then 
(i-f bx+cx i y 1 =i ‘+di \bx-\--Di \bx.cx 
tf x 2 4“ : pT~~ 1 .D^ i .ca: 3 -j- 
-f-? 4 T~\ 0 6 4 a; 4 &c. 
Hence, terms affected with a:* and x *— 1 are 
and 
-—i 7 «— i 
i ,.b 
, -2 —1 -ipil jJ! — ? » , 
C+ ? 1 •?& e + 
D ”~ 3 i“‘ . D’ b n ~ 3 C 3 + & C. 
, d £>"~ 2 c-}- &c. Now, the mth term 
