Page images
PDF
EPUB
[ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small]

The general total differential of two variables is given in terms of the general partial differentials by the formula,

d” ሪ

d” ዜ

d”
d" u

dir" + n

dir"-dy da"

dx"-dy

n (n - 1) d" u +

1.2 dir" - dy

d.xn-2 d y + &c.,

the law of the coefficients being that of Newton's Binomial Theorem.

(13 U = x"y" ;

n

d'u= m(m – 1)...(m – 3){v"-"y" dx* + 4 2*-*yn-'dx'dy

In - 3
n (n - 1)
+ 6

2017-2yu-? dx* dy
(m – 2) (m – 3)

+ 4

n (n − 1)(n 2)

20-yn-3dx dys
(m – 1)(in 2)(m - 3)
n(n − 1)(n − 2)(n − 3)

w"y"-dy'}.
m (m - 1)(m - 2)(m – 3)

+

(14) u = "+";
du = (a'd x® + 3a*bdx dy + 3 abdx dy + b3dy') eur+by.

(15) u = sin m x sin ny;
d'u = (m*dix' + 6 monod x* dy + n' dy') sin må sin ny

4mn(mod x'dy + na daw dy') cos m x cos ny.

[merged small][ocr errors][merged small]

1

du = -(a'd w? + 2ab dædy + body?)

(ax + by)** (17) 4 = (a* + y)};

1

d'u= (yo dw? 2xy dxdy + x'd yo)

(x2 + yo)

[merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors]

There is a very important theorem (due to Euler) regarding homogeneous functions of any number of variables, which from the frequent applications made of it ought to be noticed

in this place.

If u be a homogeneous algebraic function of n dimensions of r variables x, y, z, ...; then

[blocks in formation]
[ocr errors]

From this may be derived a series of equations of the form

d d Iß d
zy.
dir

dz
1.2.3...m. E

1.2.3...a.1.2.3...2.1.2.3...... = n (n - 1)...(n - m + 1) u,

(三) ()()

[ocr errors]

where a + B + y +

= ከ.

Euler, Calc. Diff. p. 188.

In applying this theorem to transcendental functions of algebraical functions, it is to be observed that it is not sufficient that these last should be homogeneous, it is also necessary that they should be of zero dimensions, as, otherwise, in the development of the transcendental function the degree of cach term would be different, and the function when expanded not homogeneous.

[merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]
[ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small]
[ocr errors]
[ocr errors]
[ocr errors]

da dy

(23) U = x (2xy + y), n = 2,

đ u + 2xy

+ y
d.x2

dy
(3 xy + 2yY) ** + 2xy (3x+ y + 3xy' + yo) – xy2

(2xy + y')#
= 2 x (2 xy + yo)

nu = x

(24) If u be a homogeneous and symmetrical function of x and y of n dimensions, so that U = X"

=y"

; and if it be expanded in terms of x so as to be of the form

2. (Qizky"-i),
then will ${(2i – n) Qi} = 0.
As u is homogeneous of n dimensions, we have

du du

+ y dx

dy' and as it is symmetrical in x and y, we have

du du

when x = y, so that dx

dy du

nu = 0 when x = y.

dx Substituting the expansion of u in this equation, we get

{(2i – n) Q; 2"} = 0, or

£{(2i – n) Qi} = 0.* This extension of a property of Laplace's Functions was communicated to me by Mr Archibald Smith.

=y

[ocr errors]
[blocks in formation]

IF y = f (x) and therefore x = f-'(y), the successive differential coefficients of y with respect to x are transformed into those of x with respect to y by means of the formulæ,

[ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][ocr errors][subsumed]

The expres

and similarly for higher orders. The reader will find the demonstration of a general formula for the change of the nth differential coefficient in a Memoir by Mr Murphy, in the Philosophical Transactions, 1837, p. 210. sion is of necessity extremely complicated, and the demonstration would not be intelligible without so much preliminary matter that I cannot insert it here, and I must therefore content myself with referring the reader to the original Memoir.

If u = f (y) and y = 0 (ir) so that u may also be considered as a function of a', the successive differential coefficients of u with respect to y may be transformed into those of u with respect to x by the formulæ

« PreviousContinue »