1360 
To each of these quantities corresponds a definite direction, viz. 
that in which we have to proceed in order to make the considered 
quantity change in positive sense while the other three remain con- 
stant. If we denote these directions by 1*, 2*, 3*, 4* and in the 
same way the directions of the coordinates #,, ,, 7,2, by 1, 2,3, 4, 
it is evident that 1* is conjugate with 2, 3 and 4, 2* with 3, 1 and 4, 
and so on; inversely 1 with 2%, 3%, 4*; 2 with 3%, 1*, 4*, and so on. 
From what has been said above about the algebraic signs of g,,, 
Jans Yow Jax it follows further that, if directions opposite to 1, 1* 
ete. are denoted by — 1, —1* etc, the directions — 1 and 1* will 
point to the same side of an extension x, = const. The same may 
be said of the directions — 2 and 2* or — 3 and 3* with respect 
to extensions 2, = const. or 2, == const, while with respect to an 
extension «,— const. the directions 4 and 4* point to the same 
side. 
Finally, we shall fix ($11) as far as is necessary, which direction 
corresponds to three others. For that purpose we shall imagine 
the directions of coordinates 1,...4 to pass into mutually conjugate 
directions, which will also be called 1,...4, by gradual changes, 
in such a way that never three of them come to lie in one plane. 
We shall agree that after this change —4 corresponds to 1, 2, 3. 
Let a,6,c,d be the numbers 1, 2, 3, 4 in an order obtained 
from the natural one by an even number of permutations. Then 
the rule of § 11 teaches us that the direction — d corresponds 
to a, b,c. It is clear that this would be the case with d, if a, b,c,d 
were obtained from 1, 2, 3, 4 by an odd number of permutations. 
If further it is kept in mind that, always in the new case, the 
directions 1%, 2*, 3*, 4* coincide with —1, —2, —3, 4, we 
come to the conclusion that the directions 1, 2, 3 and 4 correspond 
to the .sets-2*, 3% dt 3%) 14 dE 14025 A and 14) 2% JEL esses 
tively. The rule of gradual change ($ 11) involves that this holds 
also for the original case, in which 1, 2, 3, 4 were not yet mutu- 
ally conjugate. 
This is all that has to be said about tbe relations between the 
different directions. It must only be kept in mind, that whenever 
two of the first three directions are interchanged, the fourth must 
be reversed. 
§ 23. In the neighbourhood of a point P of the field-figure we 
may introduce as coordinates instead of «,,...a, the quantities 
&,...8, defined by (19). Line-elements or finite vectors can be 
resolved in the directions of these coordinates, i.e. in the directions 
