.\LIXKMKXT CHARTS IX FORKST MKNSUKATIOX TSo 



sidered to mutually intersect at an infinite distance. Now if this i)oint 

 of intersection be brought up to within a finite distance, all mutual 

 interrelations may be maintained, but the graduations of the axes will 

 obviously be distorted, since this point of intersection will now repre- 

 sent infinity on each scale. 



i>oth the efl:'ect of such a transformation and the method of perform- 

 ing it are shown in figure 8. The three light parallel lines are the axes 

 of the original addition chart which are to be replaced by the three in- 

 tersecting heavy lines. The figures at the left of the axes refer to the 

 original and those at the right to the transformed graph. The broken 

 lines indicate the method of construction by which the new axes may 

 be graduated. \\'hile in the present case the first set of these construc- 

 tion lines radiate from the zero point of the central axis, this is not an 

 essential condition and any point on this axis might have been chosen, 

 although with a difference in the resulting scale. It will be noted that 

 in the present case the values above zero are condensed and those below 

 are expanded. If the center of radiation of the construction lines had 

 been located at the original lo graduation on the central axis, the values 

 above 2 would have been condensed and those below expanded. By 

 varying the position of this center of radiation and of the point of inter- 

 section of the axes, it is evident that very considerable latitude in the 

 graduations is obtainable. It should be remarked, however, that it is 

 undesirable to attempt any material expansion of the vital portion of 

 •any scale by this plan, as all graphic inaccuracies are thereby magnified. 



In practice the preferable procedure is to first construct the parallel- 

 line form of graph, using a unit interval near the zero point which is 

 satisfactory, even though this may make a chart of excessive size. The 

 construction line from zero on the central axis to the maximum re- 

 quired value on the side axis may next be drawn. The intersection of 

 this line and the desired upper boundary of the final graph will then 

 fix the new position of the side axis. (It is, of course, not necessary 

 that the point of intersection of the axes fall within the limits of the 

 final graph.) If the higher graduations prove too condensed, the only 

 alternatives are to increase the size of the chart or to reduce the interval 

 adopted for the lower values. 



It is evident that figure 8 illustrates the method employed rather 

 than the advantages gained. Cases where the latter are prominent are 

 such as the equations ;:- = .r- -j-^'-, or tan c^ tan x -\- ta)i y, which, 

 obviouslv would be difficult or even impossible to plot for any con- 

 siderable range on parallel axes. 



