u is a function of x, y, z, X, Y, Z, and t. Therefore the equation u = 0; establishes the existence of a relation
δu = pδx + qδy + rδz = 0
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
, (6),
or,
(7),
+ X'δx + Y'δy + Z’δz;
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,
λ (pδx + qδy + rδz) = 0, or λδu = 0;
because
δu = pδx + qδy + rδz = 0.
If this be added to equation (7), it becomes
(7),
+ X'δx + Y'δy + Z’δz - λδu,
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 , which must have one direction, whose element may be represented
