24° 
Mr. Knight’s two general propositions 
+ — . z( d (p(x ' - y — ")^3 f (z/_„+w_ w y+T 1 (^— (« + 1 )+ W H«+ 
2 \ dx 2 ~~ m dy m S L 
+ ~7~( m — (h + 2) -\~ w — (n + 2)) 2 + & c - | 
+ & c - 
But by equat. (2) l n (x y) m = (x-fy — — . 
(.r +y-~ u — (» + 1 ) —■ w — (« + 1 ) ) m ' ( x ' + y ~ u — («■ + 2) — (« + 2) )”' + 
whence 
S K (o-f o') w =±{(«— « + W— «)"+ (» + 0 + W ~ (" + 0)* + 
( j.t ^ ’ ( M “-(« + 2)4* W — (« + 2)) W -j- } 
The upper or lower sign having place accordingly as m is 
even or odd. So that our equation may be expressed thus. 
<p(*>y) 
ISi2 n (o+o , ) , + 
y) — *(-£=. 
4 • <^^)*^(<>+«T+z;.s(^r>'2-(<.+0*+ 
and the analyst will, without any trouble, see the truth of 
the following rule for a function of any number of variables. 
Let the preceding values 
of 0 be — w_j, — U — 2 , — ■«_ 3, — w_ H , &c. 
of o' 6<? — w„,, — w__ 2 , — w_ 3 , . .... — w_ M , &c. 
of 0" be — — p_ 2 , — z>_ 3 , — &c. 
&c. &c. 
then will 
%*f(i x,y, z, &c.) = S n ^ +0 ' +0 " +&c ' (5) 
provided that , d/iter expansion , we multiply a term of the form 
A x «‘L(« +r) x^ ( „ +r) x t>L (M+r) x &c. 6y 
f da+b+c+Scc '<P( x > y> z > &c.> \ 
\ dx a xdy b xdz C x& c. / 
