Mr. Knight’s two general propositions, &c. 
235 
Prop. I. 
2. To find the ?2 th difference of a function of any number of vari- 
able quantities, A n cp fx, y, z, &c.), when the differences of 
x, y, z, &c. awy how variable. 
We will begin with a function of two variables ; 
Let oc —x^=u , x — x=u , x — x=u , x — x—u th , ese 
e 1 z 23 3 n n j values 
y-y=w i ,y-y=w z ,y —y=w .y- y=wy, ftX 
3 •> J neous* 
T Y , ( d H <p{x, y)\ / . . ( d U <pU, y)\ 
be represented by 2 (f^n—mfm ) ; s 'g n ^ expressing here 
the sum of all the different values that will arise to the func- 
tion within the brackets, by giving successively to m the 
values o, 1, 2, 3, n 
Lastly, let the symbol H 3 represent what may be called 
elective multiplication ; thus 2 (^~~~) ESI {u -f w) n will 
denote that each value of (~' n—mfm ) * s to multiplied by 
the corresponding term of the expanded binomial (ii-\-iv) n ; 
viz. by «, by nu- l w, (IlS-A) by 
\ dx 1 J \ dx 1 1 dyJ J \dx n - 2 d y 2 ) ^ 
n(n — 1 ) n—z . , 
- 2 u w\ and so on Then 
d^\.y+w i )==l>Q»,y)+ («+^)+T ’ 
_ £$(*, y) +w (Ab(*±yL)m u )3 . 
\dx 2 - m dy m ) 1 1 * 2 -3 \ dx 3 -tn d m) J + 
