Double and triple integrals

The differential coefficients with respect to x and y are complete. We get, on integration with respect to y, 

If the limits yx and y0 are constant, yx and y0 vanish, and we can therefore neglect the last term. If the limits also change, we must 

For the maximum-minimum condition, 817 of (82) can only vanish when the coefficient of z, namely, 

The solution of this partial differential equation furnishes z in terms of x, y, and arbitrary functions the latter must be so determined that the remaining terms of (82) vanish. 

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