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the arbitrary function being thus eliminated by one differentiation.

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60. Let there be an equation

u = ƒ {x, y, z, $, (c,), P2 (C2), $3 (C3), . . . . $m (cm)} = 0,

involving m arbitrary functions

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If we differentiate the proposed equation n times we shall have, in all, the following equations,

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the number of which is m (n + 1). We have therefore m (n + 1) functions and (n + 1) (n + 2) equations: in order to eliminate these functions it is sufficient that

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The number of quantities to be eliminated will therefore be m (n + 1) = 2m2;

and the number of equations involving them

į (n + 1) (n + 2) = ¦ . 2m (2m + 1) = 2m2 + m :

we shall therefore arrive at, as the result of our elimination, m partial differential equations between the variables x, y, z, of the order 2m

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If m = 1, then 2m 1 =

=

1; if m = 2, then 2m 1 3; if m = 3, then 2m 1 5, and so on. That is, if there be one arbitrary function in the proposed equation, there will be one final equation of the first order; if two functions, two final equations of the third order; if three functions, three final equations of the fifth order, and so on.

This is the general theory of such eliminations: it frequently happens however, for particular forms of the proposed equation, that the elimination may be effected without proceeding to so high an order of differentiation, and arriving at so many final equations, as would be implied by these general considerations. So that we must consider the general theory as defining the number of sufficient, but not in all cases the number of necessary differentiations.

Ex. 1. Eliminate the arbitrary functions from the equation

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differentiating this equation, first with regard to x and next

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Thus we see that, instead of two final equations of the third order, we have a single equation of the second order.

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The elimination of the arbitrary functions in this case can be effected only by proceeding to partial differentials of the third order, there being two final equations of this order, a result in harmony with the general theory which has been laid down. For the discussion of this example the student is referred to Lacroix, Traité du Calcul Différentiel, tom. I. p. 234.

Ex. 3. To eliminate the arbitrary functions from the equation

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The two functions are readily reduced to one: thus

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being a symbol of arbitrary functionality. Differentiating, first with regard to x and then with regard to y, we have

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This result would have been obtained by differentiating the proposed equation, without modification: the operation would however have been more tedious.

Elimination of arbitrary Functions of unknown Functions.

61. Suppose that we have two equations

u = ƒ {x, y, z, c, p(c), x(c),.. . } = 0,

....

v = F {x, y, z, c, p(c), x(c),......} = 0;

c and z being therefore implicit functions of x and y. The

functions (c), x(c),

......

are supposed to be m arbitrary

functions of c; whence it follows that c is an arbitrary function of x and y for, supposing z to be eliminated between the two equations, we shall obtain an equation between x, y, c, of which the form is arbitrary. We propose to eliminate by differentiation the m arbitrary functions of c and the function c itself.

If we differentiate the proposed equations n times successively, we shall obtain the following equations,

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the whole number of these equations, together with the two original equations, being

2 {1 + 2 + 3 + . . . + (n + 1)} = (n + 1) (n + 2).

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These (n + 1) (n + 2) equations will involve the quantities

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