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(15.) Let f(x,y)=0, be an equation cleared of irrational quan-
tities in which y is an implicit function of x, then (Art. 11.)
M +Nd.y=0,

(1) Uz + uzdzy + uz(dzy)2 + Nd;y=0, (2) and by eliminating dxy we obtain an equation of the form

M,+ Ndiy=0,
and generally an equation may be obtained of the form

Mn+ Ndry=0.
If the values of a and y which satisfy the equations

f(x,y)=0, M=0,
do not satisfy N=0, then dạy=0.
If the values of æ and y which satisfy the equations

f(x,y)=0, N=0, do not satisfy Mn=0, then dy y= o.

If f(x,y)=0, M=0, N=o be satisfied by the same values of w and y, then d.y=;, the value of which may be obtained from (2) since N=0, unless U7, U2, Uz vanish for the same values of x and y, in which case we must differentiate again, and obtain an equation of the form

V2 + v,dzy + vg (dzy)? + v. (dzy)=0; and so on until we arrive at an equation the coefficients of which do not vanish for the proposed values of w and y.

If doy=9, 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.)

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

When d,u=0, u is a maximum or minimum, according as da u is negative or positive.

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

Let the value of a which makes dau=0, make d,m+lu= (0 : U — Uy and u— U-1 may be thus developed in ascending powers of Dx;

U U-1=a;(-DX)". + a,(-Dx)"; + &c.

u-u,=a,(Dx)". + a,(Dx)", + &c. then if any of the indices m, m,, &c. have an even denominator, or if m, has an odd numerator, there is no maximum or mini

mum,

If

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

(L. C. D. 154-64.) (17.) Let u=f(x,y): the values of x and y which make du=0, d,u=0, give a maximum or minimum value of u, provided du, d, d,u, and d’u do not vanish for the same values of x and y, and that

du.d'u>(d,d,u). d'u, and diju 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 u 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 FORMULÆ.

(1.) S:du=u + const.
Saurasiu, if a is independent of x.
S:(u, + , + &c. + wn) =/;4, +S;u, + &c. +S
Sau.d.v=uv-S.v.d, u.
dou

udov S. + S.

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1

+ const. (a® + x)

a + (ao +°)3. DECOMPOSITION OF RATIONAL FRACTIONS. (4.) Every rational fraction may be reduced to the form

Ami2m + Am-12m-1 + &c. + 440 + A

2,2" + B-12- 1 + &c. + B,® + B in which m may be considered less than n.

Let a, a,, &c. an be the roots of the equation V=0; and [1] let these quantities be possible, and unequal.

If (U)r=dea V respectively represent the values of U, d, V, when w=a, then U (U)ra,

+

+ &c. + V

- 01
dra, 8 – a aq

d

.
Or thus : let V=(x a) Q, and assume

U K Р

+
V
X 01

Q'

U-KQ then

and P=

X 01 The other partial fractions may be similarly found: U– K,Q being always divisible by a —

-0,

1

(U)3=0

1

(U)r=an

1

das

V DC

x=an

K

x=,

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[2] Let the equation V=0 have r equal possible roots. U K

K

Р
Assume

+
+ &c. +

+
V (x a)" (x - a)"-1

Q

U th

d2

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1

U

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1.2

U

K,=

1 &c.=&c.

1.2...(r— 1)

U-K,Q Or thus: assume

= U,, then K,= X — a

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U,-K,Q

- a

&c.

= &c. . . .K,

U

Р

+

[3] Let the equation V = 0 have unequal impossible roots,
and let
V=(x2 +GX + e,)Q=(x-a,+Bi-1)(x-a,-B. V-1)Q,
Q not containing the factor 202 + C,& +ez.

K, +L(x+c)
Assume
V x2 + c, x te,

Q'
and let M+/-1 be the value of u {K,+L(x+ }c)}Q
when ai +Bin - 1 is substituted for x; from the equations

M=0, N=0,

U-{K,+L;(+c.)}Q K, and L, may be obtained, and P=

x2 + €, x + ei the other partial fractions, which will be of the same form, may be similarly obtained.

The values of K, and L, may sometimes be more conve

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niently obtained by substituting

.d.V for Q in the

2B. quantity U-{Ki+L1(x+c)}Q, and proceeding as above.

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