GRAPHICAL ALGEBRA. 77 



152. Case of Three or More Variables. Now let the scalar <p be a function of 

 any number of variables 



<P = f(a, & 7 ) 



In this case the discontinuous method, which consists of calculating the values of <p 

 in any sufficient number of points and subsequent tracing of the equiscalar curves 

 <p = const., may be used precisely as in the case of two variables. But if we solve with 

 respect to one of the given scalars, for instance a, in order to bring the continuous 

 method into application, we meet with the practical difficulty connected with the 

 tabulation of functions of more than two variables; for numerical tables can not 

 easily be provided with more than two arguments. 



In special cases it may be possible to decompose the complex operation into a 

 series of partial operations each depending upon two variables only. Then all 

 difficulties connected with the greater number of variables will drop out, and we 

 can bring into application the methods which we have developed already, depending 

 upon the construction of numerical tables with two arguments. 



In the general case this decomposition of the problem will not, however, be 

 possible. We must then look for other auxiliaries than numerical tables, and it 

 will always turn out to be possible to produce special graphical or mechanical 

 auxiliaries which will serve the same purpose as tables with more than two argu- 

 ments would have done. These auxiliaries will, however, as a rule be more laborious 

 to use than the tables with two variables. If, therefore, a reduction to problems 

 with two variables is possible, it should generally be performed even if the number 

 of single operations be thereby considerably increased. 



We shall give the general method for constructing graphical tables which 

 serve the purpose in the case when the number of variables is limited to three. Then 

 let a, /3, 7 be three given scalar quantities. The field of each of them is represented 

 by equiscalar curves. The problem is to find the equiscalar curves <p = const., 

 which represent the field 



(a) <P=f (a, (8, 7) 



In order to find the points of the curve a = a, in which <p has integer values, we 

 have to examine the values of 



(b) ?=/(a/3,7) 



Here only /3 and 7 are variables, and when we follow the curve a = a, (fig. 70B), we 

 see that to any value of /3 will correspond a definite value of 7, and vice versa. 



In order to find those values of one of them for which <p has integer values, we 

 construct a graphical table. We set off /3 and 7 as abscissa and as ordinate of a 

 rectangular system of coordinates (fig. 70 a) and draw in this system of coordinates 

 the curves <p = 1,2,3, according to equation (b) . We observe on the given chart 

 (fig. 70 b) the values which /3 and 7 have along the curve a = a,. These values 

 will define a certain curve in the system of coordinates /3, 7. We draw this curve 

 on a transparent sheet of paper, laid upon the graphical table fig. 70 a. This curve 



