and this must vanish by virtue of the reasoning in Art. 316; and as no relation is given between the values of dx, dy, dz,... at the two limits, the coefficients of these quantities must separately be equal to zero ;

dx

2λ dx

d2x

ds4

= 0;

and as these are particular values of the first equation of (54), it must be consistent with them; therefore a = 0; for a similar reason ẞ = 0, y = 0: whereby, and differentiating, bearing in mind that s is equicrescent,

and employing the symbols of Art. 377, Vol. I, equations (6), multiplying the preceding equations successively by x, y, z, and observing that we have

x dx+Y dy+z dz = 0,
x d3x + y d3y + z d3z = 0;

(56)

and therefore, by reason of equation (40), Art. 382, Vol. I, the radius of torsion is infinite; and therefore all points of the required curve lie in one plane.

Again, from (55), since à is an arbitrary constant, and ds is also constant, we may replace λ by X'ds: and also, replacing 1–2 by h, and kX' by h', we have

k

also because s is equicrescent,

h (x dx + y dy + z dz) = c1dx+c2dy+cądz ;

...

h(x2+ y2+z2) = 2 c1x + 2 c2y +2 Cz z + C4 j

(58)

which is the equation to a sphere: and therefore, combining (56) and (58), it follows that the curve is a plane section of a sphere, and therefore is a circle.

It may also thus be proved that the curve is a plane curve: from the last two equations of (57) we have

therefore also

h (z d2y—yd2z) = c, d2y—c2 d2z ;
.. ḥ(zdy—ydz) = c, dy—c2 dz + k1 ;
h(xdz―zdx) = c1dz−c3dx + k2,

h (ydx-xdy) = cqdx — c1dy + k ̧:

multiplying these severally by dx, dy, dz, and adding, we have

:

the equation to a plane and therefore the curve required is a plane section of a sphere.

334.] In Art. 314 it has been stated that the calculus of variations may be considered as a particular form of differential calculus, wherein the number of subject-variables of any function is infinite: I propose to illustrate this mode of viewing the calculus by the following simple example: Between two given points to draw a curve of given length, so that the area contained between it, the ordinates to the two points, and the axis of x, may be a maximum or a minimum*.

n

...

Let the two points be (x, y), (x, y): and let the distance x-x。 on the axis of a be divided into n equal parts, and the abscissæ corresponding to the points of partition be x1, x2,...X-1; and let the corresponding ordinates be y1, y2, -1; and also for convenience of notation let y1-yo = Ayo, Y2-Y1 = AY1, ... ; X1X0=AX0=x-x1 = ...; and suppose the several points, to which these coordinates refer, to be joined by straight lines, of which let the lengths be As, As,,... As-1; and let the sum of these lengths be equal to the given length c; then, if a = the required area,

C2

*For other examples of maxima and minima solved by this process see Schellbach, Variationsrechnung, Crelle, Band XLI, p. 293, 1851.

1

1 2

A = = 12 (X1−xo) (Y2+Yo) + 5 (X2−X1) (Y2 + Y1) + .....

Let the required critical function; then, à being an undetermined constant,

u = A + λ c;

and since x-x is divided into n equal parts,

X1X0= X2➖X1 = ... = a constant ;

so that u is a function of (n-1) independent variables, viz. Y1, Y2,Y-1; and therefore, taking the partial differentials of u with respect to them, and equating them to zero, we have the following series of equations:

now suppose the number of the points of division of x-x to become infinite, then, taking x, y, s to be the general types of their particular values, from (59) we have

whence by integration, a and b being arbitrary constants, we have

(x − a)2+(y—b)2 = x2.

(60)

PRICE, VOL. II.

3 P

And this is the equation to a circle. To determine a and b and

A2 sin;

(62)

(x。—a)(y,—b)—(x„—a)(y。—b) = λ2 sin

and from (61) and (62), a, b, and λ may be determined.

335.] In the preceding Article x has been supposed to be equicrescent; but we might manifestly have supposed y to be equicrescent, in which case a similar process would have led to the equation

dx dy = λd. ; ds

(63)

Also the problem might have been treated more generally; neither of the variables might have been supposed to be equicrescent; and in this case, as the coordinates would be independent of each other, we should have had two simultaneous groups of equations similar to (59); and from them, by a passage to infinitesimal subdivision, we should obtain two simultaneous equations, viz.

A careful examination of the process by which this example has been solved will shew that the method which has been employed in the previous cases, and which was explained in all its generality in the preceding Chapter, is precisely the same. In that form however it is concealed under symbols of integration and variation; whereas in this Article it has been resolved into its simplest elements, and has been laid bare to inspection and exact investigation. Other problems may of course be solved by the

same process.

SECTION 2.- On Geodesic Lines.

336.] One problem which requires the Calculus of Variations deserves especial notice; it is that of the determination of the longest or shortest line which can be drawn on a given surface from one point to another, or from one curve to another; for these lines possess important properties in the theory of geodesy, and consequently in reference to an ellipsoid of three unequal axes. The name of Geodesic lines have been given to them, and it is necessary to consider their properties at considerable length, and from various points of view.

Geodesic lines, or Geodesics, are defined to be the longest or shortest lines which can be drawn on a curved surface between two given points, or between two given curved lines.

Let the equation to the surface on which the lines are drawn be

and let us employ the same abbreviating symbols as in (1), (2), (3), of Art. 398, Vol. I. the length of the geodesic between the given limits; then s =

Let s

= ['ds; and consequently as

and as this must consist with the part of (65), which is under the sign of integration, we have

which are the differential equations to geodesic lines on a given surface: the complete integrals of them have never yet been found, but many properties may be deduced both in the general case and in the particular case of the ellipsoid.

If the geodesic line is drawn from one given point to another given point on the surface, then, as there are no variations at these limits, the definite part of (65) vanishes; but if the geo