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.

222 CALCULUS 0F VARIATIONS.
Integrating by parts,
d2c dw d( Q d) d(Q). = Qdw - -Q) +XJ +;
d2 d d d d, 2
and so for the others: it being observed, that —d, &c. signify
dx
the differential coefficients with respect to x, considering y, p, q,
&c. as functions of x. Hence, we have (ut
= + o{P- d (Q)
d t dx
+ (Q-&c.)
dx/
+ &c.
+f;,t~ d (P) d (Q)&c.}.
dx..dxi
And, putting for w, d, &c. their values Îy - p3tx, Sp - qcx,
&c.,
d (Q)
U= VrP + (0y - px.).{P- ddx + &c.}
+ (Sp- q). Q - &c.}
+&c.
___ (P d2 (Q)
+ f, N- - ( &c.}.
dx dxL
Or, since this quantity results from integration, and must, therefore, be taken between limits, if we put x, y,, p,, V, &c. for
the values of x, y, p, V, &c., at the first limit, and x, y,, p1,,
&V, &c. for those at the second limit, we have
eu a= 157% - v,~
+ ~( o y -^^) ^^^ )+& (Q,,) c.}
+ ( P,,- Av,,,n). { dx + &c.}
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