GRAPHICAL ALGEBRA. 



83 



But as long as the observations are not very good and complete it may be a 

 question if it be advisable to draw the charts for each component separately, 

 without compounding them to a vector. The formal process of drawing equiscalar 

 curves would be simple enough. But the difficulty would consist in smoothing 

 out the irregularities and filling up gaps in the observations. This must be done 

 with full understanding of the kinematical situation of which the true vector-chart 

 gives a conspicuous picture, but the two separate component-charts present only 

 a very imperfect picture. This full understanding of the situation will also be of 

 use for the control when mathematical operations are to be performed on the charts. 

 We shall therefore as a rule avoid the artificial representation of the vectors by 

 two component-fields, and use as much as possible the direct representations. 



155. Use of Angles to Represent the Directions of Vectors. We have intro- 

 duced two direct representations of the two-dimensional vector, by intensity-curves 

 and vector-lines, and by intensity-curves and isogons. We shall as a rule prefer 

 the latter when mathematical operations are to be performed. The angles which 



-E 



B A 



Fig. 75. Angles and differences of angle. 



represent the directions of the vectors A, B, . . . F will be represented by Greek 

 letters a, /3, . . . <p. We shall find it convenient occasionally in two-dimensional 

 vector-algebra to use the symbols (A, a), {B, &), . . . {F, <p) as symbols for the 

 vectors instead of A, B, . . . F. Thus we shall have identically 



(a) 



A= (A, a), B= (5,i8), 



F = (F, <p) 



In order to define completely the angles a, (3, . . . <p, we must agree upon the 

 choice of an initial direction from which they should be counted, and the direction of 

 that rotation by which they should be produced. The initial direction must be agreed 

 upon by an arbitrary choice. On our charts in horizontal projection we will choose 

 the direction toward E as this initial direction. The positive rotation around a point, 

 or, what comes to the same, the positive circidation around a closed curve, is always 

 to be defined in accordance with the positive-screw rule. Most of our charts will 

 represent fields which are contained in horizontal or quasi-horizontal surfaces. 



