552]
ON A DIFFERENTIAL FORMULA.
557
and thence
h + J (a + b) + \!{H) = i If + V(f :3 - «)) {v + v'(»r - o)j,
,.g + V(f-c)
These also follow from the known differential formula
that is,
implying
where a is a constant.
4 (da^+df) = (h-k)(~ - ^ ,
4d%dr) _ cZ/t 2 rZA; 2
V(! 2 -c)
2acZ£ cZ/i eZA?
V(f 5 - c) = V(F) + vTO ’
2eZ?7 _ eZ/i cZA;
V(#) VW
The foregoing integral formulae give at once
cZ/i _ cZ£ cZ-?7
Aff) = V(r lr Sj + VW-o)’
iZ& _ cZ| cZt?
vW)“V(P-c) _ V(^-c)’
and substituting these values we find a = 1, and the differential formulae are then
satisfied.
We thence have
const. = V{( a + h) (b + /¿)} + \/{{cl + k) (b + A;)},
as the integral of the differential equation
dh dk _
7(H) 1 j(K)~