GRAPHICAL ALGEBRA. 89 



somewhat changed arrangement as to the sign of the tabulated numbers and the 

 arguments. For the scalar value of the vector (F, <p) which is the resultant of the 

 vectors (A , a) and (B, /3) we have 



(a) F 2 = A 2 +B> 



We solve this equation with respect to one of the given quantities A or B, 



(a') B = 1 F 3 -A 2 , or A = VF*-B* 



Both formula? lead to the same table, the first of tables L, where, according to cir- 

 cumstances we can consider F and A or F and B as arguments. By this table we can 

 thus derive the intensity-curves for the vector-sum from the intensity-curves of 

 the two orthogonal vector-addends. 



In order to form the isogons of (F,<p) we have to remember that the vector 

 (5,j3) in some regions of the field may be directed along the positive and in others 

 along the negative normal to (A, a). In the two cases we shall have respectively 



/3 a =16 and a = 48 



with corresponding values of the angle <p a 



<p a<i6 and <p a>48 



The rectangular triangle will give for the determination of this angle in the two 

 cases respectively 



(b) tg O-a) = j and tg (<p-a) = -j 



We solve these equations with respect to one of the given quantities A or B, thus 



(b') B = A tg (<p-a) and B = -A tg (<p-a) 



A = B cotg (<p a) A = B cotg (<p a) 



By suitable change of arguments all formulae can be represented by one table, the 

 second of tables L. This table allows us to derive the field of the angle <p a from 

 the fields of the two tensors A and B. 



When the field of the angle <p a is found, we find by the graphical addition 



(c) <p=(<p a) + a 



the field of the angle <p which represents the direction of the vector-sum. 



The illustration of the procedure by text-figures would seem complicated, but 

 when the illuminated drawing-board is used, only four sheets of paper are required : 

 two contain the fields of the given vectors ; a third is used for the field of the auxiliary 

 quantity <p a; on the fourth we draw directly the final curves giving the fields of 

 F and of <p. 



158. Addition of Any Vectors. When the two given vectors (A, a) and (B, /3) 

 cut each other under a variable angle, the operation of determining their vector-sum 

 (F,<p) will depend upon four variables, A, a, B, 0. But the complex operation can be 

 decomposed into the following series of operations, each involving the use of two 

 variables. 



