94 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



bles, we can reduce to a smaller number of partial problems. The operations with 

 three variables which we shall have then to perform will join themselves directly to 

 those of the preceding section. 



After we have drawn the curves 



(a) (8 a = const. 



we can pass directly to the determination of the angle ip a and of the intensity F 



of the resultant by the formulae 



... 4. / \ -B sin 03- a) 



(6) tg (?-,)= A+B cqs (j8 _ ft) 



(c) F 2 = A*+B* + iAB cos (/3-a) 



In each of these formulae we can give a a certain constant value and by the 

 principles of section 152 construct a graphical table by which we can find the points 

 in which this particular curve a = const, is cut by the curves for integer values 

 of <p a and of F. We then set off A and B as abscissa and ordinate of a rectangular 

 system of coordinates, and draw in the one case the curves <p a = const., in the 

 second the curves F = const, in this system of coordinates. It will be seen at once 

 that the first curves are simply straight lines through the origin of the coordinates, 

 the second ellipse with the origin of the coordinates as center and with the axes 

 forming the angle 8 (45 ) with the axes of coordinates. It will be convenient to 

 draw both systems of curves on the same diagram. Then we can read off simul- 

 taneously the situation of the required points for integer values both of <p a and 

 of F. 



In fig. 78 we have drawn these diagrams for the values 13 a = 4, 8, 12, 16, 20, 24. 

 The radial lines <p a = const, are drawn in these diagrams for the interval 4. Thus 

 on the first diagram j8 a = 4 we have only two lines ip a = const., namely, the two 

 axes of coordinates. On the following we have 3, 4, 5, 6, and 7 of them respectively. 

 The ellipsae are drawn for unit intervals of the intensity F. The ratio of the axes 

 changes as we pass from the one diagram to the other. In the case of /3 a = 16, 

 i.e., at the curve where the vectors cut each other under right angle, the two axes 

 are equal to each other and the curves are circles. It will easily be seen that the 

 same diagrams may be used for the values 60, 56, 52, 48, 44, 40 of j3 a, taken in 

 connection with the values of <p a, which are written in brackets on the diagrams. 



By use of these diagrams, including the first of figs. 101, p. 127, we can then find 

 the points in which the curves for integer values of <p a and of F cut 14 isogons 

 13 a = const. The points of intersection with the fifteenth and the sixteenth, viz, 

 (3 a = o and /3 a = 32, have been found already by the simpler method of the pre- 

 ceding section. 



A great advantage of this method is that two draftsmen can cooperate. One 

 has before him a chart containing the three sets of curves A = const., B = const., and 

 (3 a = const. They may be copied on one paper, or they may be drawn on three 

 different papers which are placed upon each other on the illuminated drawing-board. 

 The other has the graphical table fig. 78 and a transparent paper placed upon it. 



