24 * 
in the method of differences . 
We have by form (2), 
^ n ^ x +y + +&c. +.?.+* + &c — u & c. 
i . /+3’+*+^.-«_ ( „ + 0 -r._ (n+l) -»_ ( „ +l) -& c . + 
^±0 _ /+J'+*+*c.-»_ ( „ +2) -»_ ( , 1+2) -v_ (B + 2) -&c. 
+ 
+ 
&c. 
so that 
f.. e -’«_( K+ x)- w - ( « + i)~"- (B+1 )‘- &c -+ (6) 
. ~«- ( «+ 2 )“ w -(«+ 2 r w -(«+ 2 )- &c -+ & c. 
By comparing these expressions, it appears that 
+:?+*+ &c - 
5* g°+° / +o // + & c. = JC+3>+{8 4 ' &c y and consequently, that 
£n e x+y+z+8cc. 
% n <p{x, y, z, &c. ) = - -^ +y+ ^4 &c . , provided, &c. 
It is scarcely necessary for me to observe, that the second mem- 
ber of the equation marked (6) becomes ^ e u+w+v+&:c ' \~ n 
in the case of constant differences of a?, y, z, & c. ; for u— n , 
u—( n +i)> &c. become in this case, nu, («+ 1) n, See., and the 
w, s and v, s undergo a similar change. 
The results of the preceding propositions may be brought 
into a very small compass, viz. 
A— n representing % n , the n th difference or the n th integral of a 
function of any number of variable quantities, and varying in any 
possible manner, will be expressed by the equation A n (p[x, y, z, &c.) 
_a n e x +y+ z + 8cc - 
■ x+y+K+kcT ' » provided that after expansion , we multiply 
&c. &c, &c. 
