82 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



perpendicular to each other. A vector is represented in every point of the field by 

 its components along each of the two coordinate-curves passing through the point. 

 The coordinate-curves are the vector-lines of the two vector-components. But as 

 these vector-lines are given invariable curves which are common to the components 

 of all vectors, no operations will have to be performed upon them. Although these 

 components A*, A y , B x , B y , . . . are primarily vectors, we never need take into 

 account their vector-nature. They will be represented completely by the fields of 

 their scalar values A x , A y , B x , B y , .... The sign of the scalar value will give 

 the direction of the component along the coordinate-curves. The graphical methods 

 for scalar fields which we have developed will then come directly into application 

 to all problems of vector-algebra. 



When we follow this method, the problems of graphical vector-algebra are 

 solved already. 



Thus the vector-sum F of two vectors A and B will be represented by the two 

 scalar components F x and F y , and each of them is found by graphical addition of the 

 fields of the scalar components A x and B x , respectively A y and B y , in accordance 

 with the equations 



(a) F X =A X +B X F y =A y +B y 



The scalar product of the two vectors A and B will be found by two graphical 

 multiplications and one graphical addition in accordance with the formula 



(b) A^+AyBy 



In the case of the two-dimensional fields, the vector-product of two vectors will 

 be normal to the surface which contains the field. From the point of view of two- 

 dimensional geometry it therefore loses its character of a vector. We have to deal 

 simply with a scalar 



(c) A x B y -A y B x 



and the field of this scalar is derived from those of four given scalars A x , B y , A y , B x 

 by two graphical multiplications and one graphical subtraction. 



The advantages gained by the consistent use of vector-components are great 

 enough to make it a serious question whether it should not be favorable from the 

 beginning to work exclusively with components, and not with the vectors themselves. 

 From the point of view of the observations there will be no objection against this. 

 It would be a good plan to observe separately the N.-S. and the E.-W. component 

 of the wind or of the sea-motion. If the observations were taken with self-recording 

 instruments, the vector- averages required (section 97) would be obtained by taking 

 the ordinary average of each component separately. Neither would there be any 

 objection from the point of view of the meteorological telegraphic service. Which- 

 ever system be used, two numbers will have to be telegraphed. In the one case the 

 two numbers will have to represent the two rectangular components. In the other 

 case one number must be used to represent the wind-intensity, and another to 

 represent the wind-direction. 



