1359 
we may therefore neglect the infinitesimal changes of the quantities 
Jab Over the extension considered, and also those of R, and Ry. By this 
we just come to the case considered in § 19. Thus it is evident, 
that as regards quantities of the third order the first part of (10) is 0. 
From this it follows that in reality it is at least of the fourth order. 
§ 21. Let us now return to the general case that the extension 
2 to which equation (10) refers, has finite dimensions. If by a 
surface 6 this extension is divided into two extensions 2, and &,, 
the quantities on the two sides in (10) each consist of two parts 
referring to these extensions. For the right hand side this is im- 
mediately clear and as to the quantity on the left hand side, it 
follows from the consideration that the contributions of o to the 
integrals over the boundaries of 2, and 2, are. equal with opposite 
signs. In the two cases namely we must take for N equal but 
opposite vectors. 
Also, if the extension @ is divided into an arbitrary number of 
parts, each term in (10) will be the sum of a number of integrals, 
each relating to one of these parts. 
By surfaces with the equations «,—= const.,...#,—= const. we can 
divide the extension 2 into elements which we shall denote by 
(de, . . . dr). As a rule there will be left near the surface o 
certain infinitely small extensions of a different form. From the 
preceding § it is evident that, in the calculation of the integrals, 
these latter extensions may be neglected and that only the extensions 
(de... de) have to be considered. From this we can conclude 
that equation (10) is valid for any finite extension, as soon at it holds 
for each of the elements (dr,,... de). 
$ 22. We shall now “show what equation (10) becomes for one 
element (de... de). Besides the infinitesimal quantities z,,..* x 
occurring in the equation 
FI ADN Goh La Vi == 
of the indicatrix we introduce four other quantities £,,...&,, which 
we define by 
4 
Ens Amie eS) har ow ee 3 . (18) 
or 
ES Int 3 Ti 73 le HS a ANS | 
EN ee EERDER 4°) 
S, gy as ar Jaa Va a <a ia G44 
with the equalities grq = gan. 
