Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

224 CALCULUS OF VARLATIONS.
11. The equation
d(P) d2(Q)
dx d "may, in most cases, be rendered more-easy of application
by integration. Thus, suppose there enter into V only y
and p; the equation is reduced to
d(P) d(P)
N - () =0, or Np = rp. (;
dx o r dx
d(V) dy dp d(P) dv
N - + P — =p + P
d~ dx dx dx d~'
integrating, V = Pp + C.
Suppose V involved only p and q; then the equation is
d(P) d2(Q)
reduced to - ( + (P) = 0
dx + dx =,
d(P) de(Q) d(Q)
or dx - dx2' dx
d(V) dp dq d(Q) dq
x P + Q -=q Cq + Q
dx dx dx dx d.
integrating, V = Qq + Cp + C'.
Or, suppose V contained only y and q; then
d2(Q) d2 (Q)
N+d =0, or N =
dv2 dx2
d(V) = d y d q de (Q) dq
dx dx x dx' dx
integrating, V = Qq- - + C.
And similarly in other cases.
12. We proceed to illustrate, by examples, the appli.
cation of these formula.