1355 
It is obvious that linear one-dimensional extensions can be called 
“straight lines’, also it will be clear what is to be understood by 
a “prism” (or “cylinder’). This latter is bounded by two mutually 
parallel linear three-dimensional extensions 6, and o, and by a lateral 
surface which may be extended indefinitely to both sides and in which 
mutually parallel straight lines (“generating lines”) can be drawn. 
We need not dwell upon the elementary properties of the prism. 
§ 17. A vector may now be represented by a straight line of 
finite length; the quantities X,,...Y,, which have been introduced 
in § 10, are the changes of the coordinates caused by a displace- 
ment along that line. The magnitude of the vector, expressed in 
natural units, will be denoted by JS. It is given by a formula similar 
to (1), viz. by 
SE GITE a ABW ee ok chr ie EE) 
A vector may be regarded as being the same everywhere in the 
field-figure, if \,,...X, have constant values. In the same way a 
rotation R (§ 11) may be said to be the same everywhere, if it can 
be represented by two vectors of this kind. 
If from a point P two vectors PQ and PF issue, denoted by 
Nitin Arre and XG eX | OS” resp, “the „angle “between 
them (comp. (5)) is defined by 
Gene Eb dn Mie RS dks 0A) 
We remark here that X, X;" are real, positive or negative quan- 
tities and that S’ and S" are expressed in the way indicated in (5 
“absolute” values). It is to, be understood that S does not change 
when the signs of X,,...X, are reversed at the same time. 
If S' is the value of the vector RQ and if the angle between 
this vector and RP is denoted by (S", S'"), it follows further from 
(11) and (12) that 
SPSS! cos (S', 8") + 8" cos (S", 8"). 
In the special case of a right angle R we have 
. S" = S' cos (S', 8"), 
an equation expressing the connexion between a vector PQ and its 
“projection” on a line PR. The angle (S', S") is the angle between 
the vector and its projection, both reckoned from the same point 7. 
18. Let us now return to the prism P mentioned in § 16. 
p 
From a point A, of the boundary of the “upper face” 6, we can 
