Page:Somerville Mechanism of the heavens.djvu/99

Chap II.] u is a function of x, y, z, X, Y, Z, and t. Therefore the equation u = 0; establishes the existence of a relation

between the variations δx, δy, δz, which can no longer be regarded as arbitrary; but the equation (6) subsists whether they be so or not, and may therefore be used simultaneously with δu = 0 to eliminate one; after which the other two being really arbitrary, their co-efficients must be separately zero.

In the second case; if we do not regard the forces arising from the conditions of constraint as involved in X, Y, Z, let δu = 0 be that condition, and let X', Y', Z', be the unknown forces brought into action by that condition, by which the action of X, Y, Z, is modified; then will the whole forces acting on m be X + X', Y + Y', Z + Z', and under the influence of these the particle will move as a free particle; and therefore δx, δy, δz, being any variations

or,

and this equation is independent of any particular relation between δx, δy, δz, and holds good whether they subsist or not. But the condition δx = 0 establishes a relation of the form pδx + qδy + rδz = 0,

and since this is true, it is so when multiplied by any arbitrary quantity λ; therefore,

If this be added to equation (7), it becomes

which is true whatever δx, δy, δz, or λ may be.

Now since X', Y', Z', are forces acting in the direction x, y, z, (though unknown) they may be compounded into one resultant $$\scriptstyle$$, which must have one direction, whose element may be represented