238 
Mr. Knight’s two general propositions 
, d a + b + c + &cc. ^ g> &c ) 
v. dx a . dy b . dz . &c. 
)- 
4. The expression of La Grange is a particular case of eq. 
(3), to perceive which we must observe that 
n g x + y + Z + &C.__ x + 3- + * + &c - + u n + w n + v n + &c - 
__n_ *+3' + *+&c. + « ff _ 1 +w ;f-| +w w— ,+ & c.+ 
^(w- 1 ) ^4^+g + &c ’+%_ 2 +w w _ 2 +^_ 2 + &c. , &c . whence 
1.2 1 
4 ,/+»'+<'+ to _. t “,+*« + '. +fc _l { '»-i + Vi+Vi + Sc. 
nfa—i) »„_ 2 +» II _ 1 +»»_ 1 +S*._ &c 
* 1.2 
which, if .r,y, z, &c. have constant differences, or if u n , u n ~ l , 
&c. w n , w n __ v w n _ z , &c. 27 b , w b-i &c.,&c.are»w, 
(» — 1) (» — s)w, &c., nw, (w — i)w, (w — a)w, See., nv, 
(n — i)v, ( n — z)v, &c. &c. becomes 
i g 1 J ‘ 
The equation (3) may be presented under another form; 
for if we compare the values of 
A n e and A n e we see that 
0+0 ' + o"+&c._ A n e *+y+*+&c. 
A n e 
A n dp(x,y,z,SLC.y 
,x+y+z+& c 
A n e x+y+z+Sc c. 
, consequently 
e x+y+«+Scc. 
where we must observe, with respect to the differential co- 
efficients, the same rule as was given with eq. (3). 
