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Also, using the notation of partial differentials, the above equations become

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D2u = d22u + d2u+d22u+2d,du+2d2d ̧u+2dxd ̧u+

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...

= c cos (ax+by+cz), (d)

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=c2 sin (ax+by+cz),

be sin (ax+by+cz),

D2u = — {a2 dx2 + b2 dy2 + c2 dz2 + 2 bc dy dz

+2 ca dz dx + 2 ab dx dy} sin (ax+by+cz).

In the preceding inquiry all the subject-variables of the function have been assumed to be independent, and thus the results are general. It however frequently happens that some of them are dependent on others; and although it is unnecessary to consider generally the modifications which the results undergo in such cases, yet it is expedient to give an example, so that the student may perceive the kind of results which such problems present.

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Let x be equicrescent; so that d'x=0; and from the relation given between x and y in the preceding equation of condition,

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82.] As one of the first and most useful applications of the general results of the last Article, let us prove certain properties of homogeneous functions, which are due to Euler, and are generally known by the name of Euler's Theorems of Homogeneous Functions.

DEF. A homogeneous function of many variables is one which has the sum of the indices of the variables in every term the same; and if the sum of the indices in each term = n, the function is said to be homogeneous of n dimensions. Thus

ax3 + bx2y + cz3 + exyz + gx2z = 0

is a homogeneous function of three dimensions.

Let u = F(x, y, z, ...) be a homogeneous function of n dimensions and r variables; for x, y, z,... let tx, ty, tz, let tx, ty, tz, ... be written, and suppose the function to become u when these substitutions are made; then, by the definition of homogeneous functions,

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... u' = v(x', y', z', ......);

and therefore by (46), Art. 49,

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equating this to (110), because they are equal, we have

(111)

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let t1, then xx, y=y, z=2, ...;

.'. n F(x, y, z,...) = NU=X

dx

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Again, taking the second total differential of u', and dividing

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whence, equating (113) and (114), and making t = 1, we have

;) + y2 (d2u)

n (n-1) u = x2

(du) + 2xy (dx dy

d2u

d3u

dy2

+ .. (115)

dx2

Similarly it may be shewn that

d3u

n (n − 1) (n − 2) u = x3 (d/34 ) + 3x3y (dx3 dy

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and similar theorems are true for every other order of differentials. One or two examples are subjoined, in which the theorems are shewn to be true.

Ex. 1. F(x, y, z) = u =

x2+By2+cz2+EYZ+GZX+HXу,

which is an homogeneous function of two dimensions and three variables.

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Ex. 2. u = F

dimensions.

(du) + 2y z (

d2u

d2u

dy dz

(d2) + 2xy

dz dx

d2u

(

dx dy)

2 {Ax2 + By2+cz2 + Eyz+Gzx+Hxy}

=

= 2u.

(2), which is a homogeneous function of O

Of F(2) let F'(2) be the derived-function, that is, be that quantity which multiplied by dz is the z-differential of F(2); then

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Article 53, Ex. 4, is another case in which the preceding theorem is exhibited.

83.] As another application of the results of Article 81, let us investigate the equivalents of the second derived-functions of a function of three variables, when one, say z, is an explicit function of the other two, in terms of the derived-functions, when all are implicitly involved in an equation. That is, if z = f(x, y) and p(x, y, z) = 0 are equivalent equations, it is red2z

quired to express (4), (dry), (d) in terms of (),.

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therefore if we estimate the simultaneous variations of x and z,

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let the x- and z-differential of this be taken, and let a be equicrescent; then we have

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therefore dividing through by de2, and replacing (dz) by its

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dx dz

is expressed in terms of the partial derived of

Similarly for the y- and z-partial differential

2

(172) (dt) (12) + (124) (dg)2 + (de)' (d)

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dz

dy dz dy

dz

αφ dz2 dy dz

= 0.

Also taking the y- and z-partial differential of (118), and sub

dy), as they are given in Art. 50, we

dz

dz

stituting for

dx

), and

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We proceed now to consider the theory of successive derivation of an implicit function in its application to many and important theorems.

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