in the method of differences. 
*39 
Prop. II. 
5. To find % n <p(x, y, z, &c.) supposing that the differences of 
x, y, z, &c. are any how variable . 
We shall here make use of the same notation as we did 
in Prop. I, only let the preceding values 
of x be x —u_i, x—-u_ z , x — w_ 3 , &c. 
of y bey — w„i,y — w_ 2 ,y-— m__ 3 , &c. 
of % be z—v—u z—v_ z , z — v^ 3 , &c. 
&c. &c. 
It will be sufficient also to consider the case of two variable 
quantities, as was done in Prop. L 
First, we have, in general, 
I p(x—U_ r , y— W_ r )=(f> (x, y)—t( ~~~ ] gl(«- r +W-r)+ 
\dx dy J 
+ 4 . is lu_r + »_ r y - 4 . 
2 W-” iy m ) K T 1 z ' 3 W-V/ 
ESI (w.„ r + w —/-) 3 + which expression being combined with (2), 
putting for the sake of symmetry a? (.r, y) = 2 ( - ^ Y 
\dx° m dy m J 
gives 
£” f(x, y) = £ (f flfp ) El { («-»+«'-»)”+ T •(«-(»+.) 
*4 (» + 2) + a ’— (• + «))”+ ^ C - } 
'dx dy / 
+»_<„+, ) ) , + : rr(«-(. +I )+w-( l +:)) , + } 
I i 
MDCCCXVII. 
