\ 
242 Mr. Knight’s two general propositions 
6 . Scholium. 
We may find, in many cases, very elegant and regular 
expressions for A n <p(x), by supposing <p(x-\-u ) to be expanded 
differently from the form given by Taylor’s theorem: as, 
for instance, 
ijf f (x + u) = 4,( x )+X.a:(u)+X' x '(u)+XV(#)+.(7) 
where X, X', X", &c. represent any functions whatever of x, and 
<p, X’ X 1 &c. a ny functions of the quantities they stand before , 
then A x, or u, being constant, 
A«?>(a:)=X. A». % (o)4-X / .A w .% , ( 0 )-F X/ '- Aw -% ,, (°H- &c - : far 
?>(# + nu)=y \ (a?) + X .% (nu) -f X'. %\nu) -f 
— ^-.(p(x-\-n - — 1 ,u)—-~ ~ . — -~X.x(n~i . u ) — ~ .X'. 
x\n — 1 . u ) — 
+’■^•+5=5 ■ • X . &=i . ,)+ 
^.X'. X '(— 2 ).«)+ 
&C. &C. 
and because A”. <p=*p — * <P w __ 2 — , this being 
added give 
A”. $(x)=X . A”. % 0 )+X'. A*. % '(o)+X". A”. z "(o)+ (8 ) 
If form (7) soon terminates, the expressions for the diffe- 
rences are very simple, as in 
Esc. 1. Sin. (a? -{- u ) — sin. a? cos. w + cos. a?, sin. z/, which 
being compared with (7) gives X == sin. a?, X / =: cos. a?, X", 
&c.=o ; x(o) — cos. 0, %' (0) = sin. 0, %" (o), &c. = 0; whence 
A w . sin. a? = sin. a?. A”, cos. 0 + cos. a?. A w . sin. 0. 
I£r. 2. Tang, (a? tang, a? + sec.*a?J tang. M-j-tang. x. 
