April 21, 1916] 



SCIENCE 



579 



when a is negative and c is very large, we 

 may have two real roots; and there are other 

 portions of the field where three real roots 

 occur. 



A use of the diagram, apart from the solving 

 of equations, is thus to indicate the niunber of 

 real roots that exist in the case of particular 

 values of a, h and c. It is apparent that the 

 chart will also indicate the effect of changes 

 in the magnitude of the coefficients a, b and c 

 on the absolute value or sign of x; and the 

 reader will perceive that transcendental equa- 

 tions beyond the range of ordinary algebraic 

 methods can be solved by this means. 



Fig. 3. 



A further use for a chart of this kind is to 

 suggest a proper empirical equation for the 

 representation of experimental results. Thus, 

 if the data collected in a series of experiments 

 are believed to be expressible by an equation of 

 the form y = ax -\- bx'^, the chart given in 

 Fig. 2 may be used to determine the proper 

 value of the coefficients a and b. The details 

 of this procedure hardly require explanation; 

 and other diagrams have already been pub- 

 lished that constitute a graphical substitute 

 for the method of least squares. 



Returning now to the general case, it is 

 evident that if F(x) =■ 0, we have 



in this case the scale for F(x) shrinks to a 

 point at zero, through which all the lines repre- 

 senting different values of x must pass. 



If f(x) and F(x) are constants the general 

 e<iuation takes the form 



af(w) +lf{v)=f(y). 



This may be charted as the so-called " aline- 

 i7ient chart," well known to students of graph- 

 ical mathematics.^ 



If F(x) is replaced by /(z) in the general 

 equation we have five variables to consider. In 

 this case f(z) is plotted along the right-hand 

 axis, the series of lines marked with the differ- 

 ent values of x being omitted from the chart 

 when the latter is first constructed. To use 

 such a modified chart locate the point D in the 

 usual way, then pass a straight line through 

 D and that point on the right axis marked 

 with the given value of z. The point of inter- 

 section of this line or its prolongation with 

 the left axis gives the required value of x. 



If two equations be given in which the 

 values of x and z are to be determined, we 

 locate two points D and D' in the usual way 

 from the given values of u, v and y. Draw a 

 straight line passing through D and D'. Its 

 intersections with the left and right axes will 

 give the values of x and z which simultane- 

 ously satisfy the equations. A set of three 

 simultaneous equations of the general form 



f(u) ■f(x)+f(v) .f(z)+f{y)=0 



may be solved by an extension of this method. 



It will be noticed that in the ease last con- 

 sidered we are treating five variables, instead 

 of the three that are included in the ordinary 

 alinement chart. It was, indeed, by an exten- 

 sion of the principles of the alinement chart 

 that the method presented in this paper was 

 devised. 



Exponential equations of all sorts may be 

 handled by this method. Thus 



2 See, for example, Peddle, ' ' The Construction of 

 Graphical Charts," New York, McGraw-Hill Book 

 Co. 



