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derivatives and we shall determine the variation it undergoes by arbitrarily chosen variations , these latter being continuous functions of the coordinates. We have evidently

By means of the equations


this may be decomposed into two parts




The last equation shows that


if the variations and their first derivatives vanish at the boundary of the domain of integration.

§ 35. Equations of the same form may also be found if is expressed in one of the two other ways mentioned in § 33. If e.g. we work with the quantities we shall find

where and are directly found from (43) and (44) by replacing , , , and etc. by , etc. If the variations chosen in the two cases correspond to each other we shall have of course

Moreover we can show that the equalities

exist separately.[1]

  1. Suppose that at the boundary of the domain of integration and . Then we have also and , so that

    and from