1358 
> 
and X, do in the first of the integrals (16) annul each other. lt 
will be clear now that the whole integral vanishes and that similar 
considerations may be applied to the other three. 
So we have proved that under the special assumptions made the 
left hand side of (10) will vanish in the special case that the directions 
of the coordinates are perpendicular to each other. This conclusion 
likewise holds for an other set of coordinates if only the assumption 
made at the beginning of this § is fulfilled. This is obvious, as we 
can pass from mutually perpendicular coordinates v,,...a, to arbi- 
trarily chosen other ones 2’,,...2’, which fulfil this latter condition 
by linear transformation formulae with constant coefficients. The 
w- and the w2’-components of the vector 
[Re . N] Le [Ra 3 N | 
are then connected by homogeneous linear formulae with coefficients 
which have the same value at all points of the surface 6. Hence if, 
as has been shown above, the four w-components of the vector 
» e 
fi [Re . NJ + [Ra . N]}do 
vanish, the four 2’-components are now seen to do so likewise. *) 
§ 20. The above considerations were intended to prepare a 
corollary which will be of use in the treatment of the integral on 
the left hand side of (10), if we now leave the special assumptions made 
above and suppose the quantities gq, to be functions of the coordi- 
nates while also the rotations R, and R; may change from point 
to point. 
This corollary may be formulated as follows: If all dimensions 
of the limiting surface » are infinitely small of the first order, the 
integral | 
fou NT + IRA. NI fed 
will be of the fourth order. 
In order to make this clear let us suppose that in the calculation 
of the integral we confine ourselves to quantities of the third order. 
The surface ó being already of that order we may then omit all 
infinitesimal values in the quantities by which Wo is multiplied; 
1) In the above considerations difficulties might arise if the vector N lay on the 
asymptotic cone of the indicatrix, our definition of a vector of the value 1 would 
then fail (comp. note 2, p. 1345). With a view to this we can choose the for mof 
the extension © (§ 13) in such a way that this case does not occur, a restriction 
leading to a boundary with sharp edges. 
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