A Synopsis of the Principal Formulae and Results of Pure Mathematics

(15.) Let f(x,y)=0, be an equation cleared of irrational quantities in which y is an implicit function of x, then (Art. 11.)

M+Nd,y=0,

u1+u2d ̧y+u ̧(d ̧y)2+Nd2y=0,

(1)

(2)

and by eliminating day we obtain an equation of the form

M2+Nd2y=0,

and generally an equation may be obtained of the form

M2+Nday=0.

If the values of x and y which satisfy the equations

f(x,y) =0,

do not satisfy N=0, then dry=0.

M=0,

If the values of x and y which satisfy the equations

do not satisfy M1=0, then dy = ∞ .

If f(x,y) =0, M=0, N=0 be satisfied by the same values

from (2) since N=0, unless u1, u, u, vanish for the same values of x and y, in which case we must differentiate again, and obtain an equation of the form

v1 + v2 d2y + v ̧ (d ̧y)2 + v4 (d ̧y)3 =0;

and so on until we arrive at an equation the coefficients of which do not vanish for the proposed values of x and y.

If dry=-, its values may be similarly found.

If for any system of values of the variables, any differential coefficient has more than one value, then that coefficient obtained from the differential equation of the corresponding order must appear under the form

(L. C. D. 135—7; L. D. C. 109—13.)

(16.) Let the values of u corresponding to a + Dx, a−Dx be u1, u-1 respectively; then if u-u, and u-u, have the same sign for all values of a from a to a+ Dx, the value of u when a = a is a maximum or minimum, according as u-u, and u-u-1 are both positive or both negative.

When du=0, u is a maximum or minimum, according as dau is negative or positive.

If the same value of a makes du =0, there is no maximum or minimum unless du also vanishes: and generally there is no maximum or minimum, unless the first differential coefficient that does vanish with any given value of x is of an even order,

Let the value of a which makes du =0, make dm+1u= ∞ : u-u1 and u-u-1 may be thus developed in ascending powers of Dx;

u―u_1 =α1(—Dx) + α2(−Dx)"2+ &c.

u—u1 = a1(Dx)11 + α2(Dx)2 + &c.

then if any of the indices m, m2, &c. have an even denominator, or if has an odd numerator, there is no maximum or mini

mum.

If m1 has an even numerator, u will be a maximum or minimum, according as a, is positive or negative.

(L. C. D. 15464.) (17.) Let u=f(x,y): the values of x and y which make du=0, du=0, give a maximum or minimum value of u, provided du, dedu, and d'u do not vanish for the same values of x and y, and that

du.du>(dd,u).

du, and džu must evidently have the same sign: and generally there will be no maximum or minimum, unless the first series of differential coefficients which do not vanish for any given values of x and y be of an even order; and unless the series of vanishing differential coefficients might be the coefficients of an equation of the same dimensions, which has no possible root.

(Garnier, Diff. Calc. Ch. xxv.)

INTEGRAL CALCULUS.

FUNDAMENTAL FORMULE.

(1.) du=u+const.

fau=afu, if a is independent of x.

ƒ.(u2+u2+ &c. + u„) =ƒ‚μ‚ +ƒ‚μ2+ &c. +S2u‚„• fu.d.v=uv-fv.du.