1561 
1%, 2%, 35, 4* Their components and the magnitudes of different 
extensions can now be expressed in §-units in the same way as 
formerly in v-units. So the voiume of a three-dimensional parallele- 
piped with the positive edges dE, d&,, dE, is represented by the 
product d&, dé, dé,. 
Solving a,,...«, from (19) we obtain expressions of the form 
De Nay EMEP Ta Sars oe Yaa Se | 
MIP ea URINE en, ies EED (20) 
i PS Nite or tk tte Heat ey 
Vla = Yab 
If we use the coordinates £ the coefficients y‚: play the same 
part as the coefficients gu, when the coordinates w are used. According 
to (18) and (20) we have namely 
Bi (a) Er (0b), vig Sa Shs 
so that the equation of the indicatrix may be written 
(GO) Faq Sa Sh = 87: 
§ 24. Let the rotations R, and R; of which we spoke in § 13 
be defined by the vectors Al, All and All, AlV respectively, the 
resultants of the vectors Arl,... Aal, ete. in the directions 1%... 4%. 
Then, according to the properties of the veetor product that were 
discussed in $ 11, 
[RoN] = [(Asel +... 7 -b- Aas!) (Ari H.+ Auel). NJ 
= XE (ab) { [Aasl . Ase TE. NJ — [Agel , Apel. NJ}, 
where the stroke over ab indicates that each combination of two 
different numbers a, contributes one term to the sum. For the 
vector product [R‚.N] we have a similar equation. Now two or 
more rotations in one and the same plane, e.g. in the plane a*d*, 
may be replaced by one rotation, which can be represented by 
means of two vectors with arbitrarily chosen directions in that plane, 
e.g. the directions a* and 5% We may therefore introduce two 
vectors Ba» and Bj+ directed along a* and 5* resp., so that 
[Bax . Bor] == [Ast Ave] — [Agel . Agel] + 
Ae LE Apel VSS [AnelY (Apel) 421) 
Then we must substitute in (10) 
[RNTC PN] S= (ob) [Babe N] ©. 7. (23) 
Here it must be remarked that the magnitude and the sense of 
one of the vectors B may be chosen arbitrarily ; when this has been 
done, the other vector is perfectly determined. 
