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.

CALCULUS OF VARIATIONS.
1. IN solving problems of maxima and minima by the
Differential Calculus, it is necessary to express the quantity,
which is to be made a maximum or minimum, in terms of
the independent variable, or at least to find an equation
between them; in all cases it is necessary to assign a relation
between the function (uz) and the independent variable (x),
so that for any given value of x, the corresponding value
of u actually can be found. When this is the case,
dG
can be found; and if we make it = 0, an equation is obtained, which, combined if necessary with the original equation, determines the value of x or u, corresponding to the
maximum or minimum value of u; and the problem is
solved.
2. But it is sometimes necessary to solve problems of
maxima and minima, when the relation between u and x
cannot be expressed, u depending generally upon an integral
FV, where V involves y and its differential coefficients, and
where it is the object of the problem to find the relation
between y and v. Suppose, for instance, it were required
to find the curve of quickest descent from one given point
to another: then, if m be measured horizontally from the
first point, and y vertically, and if the time =u, we have
du v ~cy^W f (d/(l+ 2)
di~ /(2gy) ' u= / (2gy)
dy
where p d
dx
This expression plainly cannot be integrated except the
relation of y to o, be given; but this it is the object of
the problem to discover. Here then the methods of the