90 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



(i) By graphical subtraction we form the auxiliary field of the scalar /3 a 

 which represents the angle between the two given vectors. 



(2) By graphical division (section 151) we form the auxiliary field representing 



the ratio -r of the numerical values of the two given vectors. 



(3) By the elementary properties of the triangle with the sides A , B, and F we 



get the following relation connecting the angle <p a with the known angle a and 



. B 

 the known ratio -r 



T> 



We solve this equation with respect to -r and tabulate this quantity as function 



of the two angles /3 a and <p a. Using this table, the first of tables M, we can 

 derive the field of the angle <p a from the fields of the two auxiliary quantities 



T> 



j8 a and -r. 



A 



(4) By the properties of the same triangle we find the following relation which 



F B 



connects the ratio -7 with the ratio -r and the angle fi a, 



() *= + '+! *.-) 



We solve this equation with respect to -7 and tabulate this quantity with the ratio -j 



and the angle /3 a as arguments. This gives the second of tables M. Using this 



F 

 table we can derive the field of the ratio -j from the fields of the two auxiliary quan- 



tities /3 a and -j. When two numbers are given in the same place in the table, 



the curve -7 = const, has two points of intersection with the curve /? a = const. 



Both will have to be used. 



(5) By graphical multiplication (section 150) we derive the field of the intensity 



p 



F of the required vector from the fields of the ratio -7 and of the intensity A of the 



given vector. 



(6) By graphical addition we derive the field of the angle <p of the required 

 vector from the fields of the angles <p a and a. 



It should be emphasized that as soon as we have drawn the first two systems 

 of auxiliary curves (1) and (2), we know the situation of all zero-points of the field. 



Every singular point of a vector is a zero-point for its absolute value. The 

 resultant can be zero only in points where the two given vectors have equal magnitude 

 and opposite direction. Now the two given vectors have equal magnitude in the 



Tf 



points of the curve -r = 1 , and opposite direction in the points of the curve j3 a = 32 . 



