1350 
a vector may be resolved into four components which have the 
directions of the coordinates, viz. such directions that a shift along 
the first e.g. changes only v,, while v,, 7,, «, remain constant. The 
four components in question -are determined by the differentials 
dv,,..dv, corresponding to PQ. We shall say that by these they 
are expressed in ‘\v-measure’. Their values in natural measure are 
found by multiplying de,,. .de, by certain factors. If we keep in 
mind that the radius-vectors of the conjugate indicatrix and the 
indicatrix in the directions of the axes are expressed in “wv measure” by 
é . é € € 
Vo, 
’ = 9 
V—g,, Vv —4g,, Vg 
and in natural units by 
’ 
16. VE, 26,52 © 
we find for the reducing factors 
et) aa ee =| Ef Sik a ae L—=V4q,, : (7) 
In the language of vector-analysis the vector obtained by the 
composition of two or more vectors is also called the sum of these 
vectors. . 
We shall also speak of jizte vectors, i.e. of directed quantities 
which can be represented on an infinitely reduced scale by line- 
elements in the field-figure. If w is the constant “reduction factor” 
chosen for this purpose, a vector A will be represented by a line- 
element wA, the direction of which is also ascribed to A. ‘It will 
now be evident that two finite vectors, as well as two infinitely small 
ones, determine an infinitesimal two-dimensional extension and that 
finite vectors can be compounded and resolved by means of parallelo- 
grams and parallelepipeds. Also that we may speak of the “magnitude” 
of such figures, that e.g. the rule given in § 8 applies to the parallelo- 
gram described on two vectors. 
The components of a vector in the directions of the coordinates 
expressed in v-measure will be called X,, X,, X,, ,. This means 
that wX,,...wX, are equal to the differentials dz, ,...de, cor- 
responding to the infinitely small vector wA. 
If we want to know the components of A in natural units we 
must multiply X,,...X, by the factors (7). 
§ 11. Two vectors A and B starting from a point P of the field- 
figure and lying in a plane VV, determine what we shall call a 
rotation R in that plane. We ascribe to it the direction indicated by 
the order AB and a value given by the parallelogram described on 
