( 715 3 
§ 4. If we have to do with a set of vector-quantities of one kind 
or another — but all of the same kind — cach belonging to one of 
the material points, we shall call the complex of all these quantities 
a vector in the system or simply a vector. The rectangular compo- 
nents of the several vector quantities will be called the elements of 
the vector in the system. 
From this it follows that an infinitely small displacement is itself 
a vector in the system, and that any vector may be geometrically 
represented on an infinitely small scale by such a displacement. The 
length or value of a vector and the angle between two vectors may 
be defined in a similar way as the corresponding quantities in the case 
of infinitely small displacements. 
We shall often denote a vector by the letter S, its value by S, 
its elements by X,. Accents or other suffixes will serve to distinguish 
one vector from another. Other Gothic letters for vectors, and the 
corresponding Latin ones for their values will likewise be used. If 
an infinitely small displacement is to be regarded as a vector, we 
shall denote it by ds or 08. 
The value S of a vector, considered in most cases as a positive 
quantity, is given by the formula 
3n 
mS = N; mX ee aa TANT Ot Me ae (3) 
1 
and the angle (©, ©) between two vectors by 
3n 
TSS cos (Sy Sy == Sy An XE i ee eae eed 
(S, ©) pe (4) 
If (SS) =0, the vectors are said to have the same direction. 
For this it is necessary and sufficient that the ratios between the 
elements X, should be the same as those between the elements X,. 
The ratios between the elements and the length will then likewise 
be the same for the two vectors. It is natural to call these last 
ratios the direction-constants. If these are @,, so that 
ay mg 
S 
the equation (4) becomes 
3n 
m cos (S, S') = Ww my ary ce. NER ed (5) 
1 
The angle between two vectors depends therefore on their direction- 
constants, or, as we may say, on their directions. 
The direction-constants of a vector may not be chosen indepen- 
dently from each another, the relation 
