7SiS JOURNAL OF FORKSTKV 



In some cases the original chart becomes prohibitively large when 

 this method is attempted, and it is necessary to find an algebraic expres- 

 sion by means of which the new axis can be graduated. Where the 

 center of radiation of the construction lines is at the zero point of the 



central axis, as in figure 8, this is u= ^— J— - , where // is the vertical 



distance above the zero point of any graduation on the new axis, v the 

 vertical distance above the zero point for the corresponding graduation 

 on the old axis, and c the vertical distance above the zero point of the 

 point of intersection of the axes. In figure 8. for example, if the 

 original scale of the central axis be employed the graduations on the 



sloping axis may be found at — "^ — units above the zero point. For 



the 6 graduation on the left-hand vertical axis, which is 12 units above 



the zero point. 't'=T2, and ^=6; it will be noted that this 



^ 12 -|- ^' 



graduation on the inclined axes lies at this height above the base. 



In such cases as are illustrated by the equation ■■\ T = '\ x + N '/ 

 the graduations on the simple form of chart become rapidly too con- 

 densed. This condition can be remedied by placing the common point 

 of intersection on the negative side. Here, since the graphic method in- 

 volves the expansion of small intervals, and hence a serious source of 

 error, as has been already noted, it is better to calculate the position of 

 the graduations on the outer axis, after which the central scale can be 

 prepared by intersections. The formula in this case is 



cv 



In an entirely similar way one of the parallel lines of a .;; chart can 

 be swung with its intersection with the diagonal as a pivot. The con- 

 struction in this case is parallel to that just described, and the only 

 material difference in the result is that while one of the scales is con- 

 tracted in its useful portion, the other is expanded. This statement 

 applies to the two axes which were originally parallel, the diagonal 

 being entirely unaffected by the transformation. The same formula as 

 in the previous case may be used for locating the graduations on the 

 altered axes, if u is understood to represent the distance measured 

 vertically above the diagonal axis. 



Figure 9 illustrates a simple rr chart thus altered. In this example 

 the point of intersection is beyond the limits of the chart. Construction 

 lines are indicated, as before, by broken lines. 



