GRAPHICAL ALGEBRA. 



73 



Instead of following the vertical columns we can also follow the horizontal lines 



of the table, and then draw directly one by one the curves <p = const., performing 



successively the interpolations by eye-measure which give the points of intersection 



with the different curves a = const. This method will usually be the most convenient. 



Tables G. Table-schemes for graphical algebra with two variables. 



The second table G can be used in precisely the same way to find the points of 

 intersection of the required curves <p = const, with the given curves (3 = const. 

 When <p is a symmetric function of a and 0, the two tables will be identical with 

 each other. Then one table will be sufficient, which may be provided with two 

 sets of arguments, one set above and on the left side, the other below and on the 

 right side. (Cf. tables H and I below). 



We shall now make a few special applications of this general principle, taking 

 the simplest algebraical operations, and giving the schemes for the construction of 

 the most important auxiliary tables. More extensive tables will be given later in our 

 collections of tables for practical use. 



149. Addition of Scalar Fields. Let the function be / (a, /3) = a+/3. That is, 

 we shall determine the field of the scalar <p which is the sum of the scalars a and 13 



(a) <p = a+ /3 



The discontinuous method will consist in forming directly the sum a +^3 in a certain 

 number of points, and to draw the equiscalar curves of <p by leading of these values. 

 In order to use the continuous method we write equation (a) in the form 



(b) (i = <p a or a = <p fi 



Both equations lead to the same table, table H, where on account of the symmetry 

 we have an equal right to interpret a as argument and as the tabulated quantity 

 or @ as argument and a as tabulated quantity. 



The table shows that the curves representing the sum of the scalars a and pass 

 through the points for simultaneously integer values both of a and as a set of 

 diagonal curves (figs. 66 and 67), i. e., we return to the simple process of graphical 

 addition, of which we have made so frequent use. In this simple case the auxiliary 

 table is superfluous. We have introduced it only to show the connection with the 

 more complicated corresponding problems. 



It will be seen at once that while the sum a+/3 is represented by the one set of 

 diagonal curves, the difference /3 a or a will be represented by the other set 

 of diagonal curves. 







