g is the minimum, and c the maximum, value of x: beyond these limits ≈ is impossible.

y is a maximum, when a={(c2+go)]3.

[9] :

In this case the curve is a circle.

(Whewell, Mech. Appendix.)

CALCULUS OF VARIATIONS.

For a history of the origin and principles of the calculus of variations, see (Woodhouse's Isoper. Prob.; L. C. D. 825-43.)

from which du may be obtained in terms of dy and dx.

(2.) dƒx(x,y,Y',y", &c.) =df2 V

=VSx+ {d ̧V — d ̧d, V+ &c. }(dy—y'dx)

+ {d, V― &c.} d ̧ (dy—y'dx)

+ &c.

(1)

+f{d,V-d ̧d, V+ d2d, V- &c.} (dy-y'da) (2)

In order that V may be integrable with respect to a, we must have d1V-ddV+ dd," V — &c. = 0.

From this last equation it follows that the quantities

Sady V,

S {Sady V-dy, V},

Sz{S (Sdy V — d„,V)+d„„V}, &c.

may always be determined.

(L. D. C. 456-67; L. C. D. 844-53.)

(3.) If V=${x,y,y'‚y", &c. ƒ‚†(x,y,y', &c.)} = p(x,y,y',y", &c. f, V1),

let dy-y'dx=w, S.V1=u, and f,d, V=U, then SSV=Vdx + {d ̧ V — d ̧d, V + d2d ̧„V — &c.} w

+ {d, "V — d ̧d,V + &c. } d2w

- S2{Ud, V2- d2 Ud ̧ V2+d, Ud, V1— &c.}w.

(4.) Let V={x,y,x, p, q, r, s, t,...&c.},

In which p,

represent

q, r,

(L. C. D. 854.)

d, d ̧é, d2x, ddx, d2≈, d3x, di̟d≈, dd2x, d3x,

respectively, and let w=dz-pdx-qdy, then

SxSyV=w{d ̧V— d ̧d, V+ &c.)

-dydu V+ &c. f

+d,w{d, V- &c.}+&c.

+dow {d1 V- &c.} + &c.

+S, Vdx + S2 Vdy

MAXIMA AND MINIMA OF INDETERMINATE INTEGRALS.

(5.) Since the part (1) of the value of

V has been obtained by integration, it must be taken between limits: let the values of V, x, y, y'; &c. corresponding to the first limit be V1, x1, Y1, y'15 &c.

19

and to the second v2, x2, Ye, Y'2; &c. then

+ {dy, V2 − dx, dy, V2 + &c. } (dy2 — y'2dx)

2

+fx{d„V—d ̧d,,V+d2d,, V—&c.} (dy—y'dx).

(2)

(6.) If a maximum or minimum value of V be required, then V=0, and hence the quantities (2), (3) must be separately equal to nothing. Eliminating therefore between (3) and any given conditional equations as many as possible of the quantities Sx1, Sx2, Sy1, Sy, &c., and making the coefficients of the rest severally =0, we may obtain the value required.

The required result may frequently be more easily obtained from the condition

d1V — d2d, V+ d2 d„„V — &c. =0.

(7.) If in addition to the condition U1=maximum or minimum, we have also U=0, then 8U=0. Both these conditions may be expressed by the equation

2

ques

a being a constant to be determined by the nature of the tion. From this equation da, da, &c. must be eliminated, as in the preceding case.

Similarly if the condition U ̧=0 be also given, then

3

and so on for any number of independent conditional equations.

(Airy, Math. Tracts)