hy Means of Graphical Methods. 265 



angle from x' intersects the vertical line at a point c, which, 

 when measured with scale No. tt of the printed sheets, gives 

 the length of the vertical axis, c=0-66. The whole procedure 

 of determining the lengths of the a and the g axes is exceed- 

 ingly simple both in its principles and in its execution, and 

 with slight modifications may be applied even to the mono- 

 clinic system where the h axis is at right angles to the a and g 

 axes. 



Instead of laying off the hase-line Ox' to the right of the 

 center by means of dividers, as shown in figure 15, it may be 

 plotted to the left, by making use of the principle that Ox 

 equals the sine of r. Accordingly locate m to the left on the 

 divided circle, and draw a vertical chord ; then plot the slope 

 of p from the center, making use of the graduation of the 

 divided circle, and measure tlie length of the vertical axis on 

 the chord x'g. On one divided circle, therefore, a stereo- 

 graphic projection may be made in three quadrants, reserving 

 the upper left hand quadrant for plotting the length of the 

 vertical axis. It may be noted that almost at a glance at such 

 a chart the two important angles of the two-circle goniometer, 

 ip and |0, may be told from the graduation ; for example, the 

 radius through^, figure 15, determines m, and, therefore, the 

 angle ^, measured from 010, which is nearly 51° in figure 15 ; 

 and the slope of p, plotted from the center and intersecting 

 the vertical chord at 0*66, meets the divided circle at a little 

 over 46°, which is the angle ^o, measured from 010. For 

 plotting measurements made with the two-circle goniometer 

 and interpreting the results, the method is most admirably 

 adapted. 



Having explained at considerable length the principles in- 

 volved in solving problems graphically, some examples of the 

 application of the methods to specific problems in the several 

 crystal systems will next be given. The examples cited are 

 problems which have been encountered in the writer's labor- 

 atory, and in all cases it will be assumed that the fundamental 

 measurements were made to the nearest minute, hence the 

 results of plotting will indicate the accuracy of the graphical 

 methods. Generally, the scheme indicated in figure 15 will be 

 followed, that is, the stereographic projection w^ill be made on 

 the lower half of the divided circles, reserving the upper half, 

 or the upper left hand quadrant, for plotting the intercepts on 

 the vertical axis. The horizontal radius, from the center to 

 the right, will indicate tlie h axis and the lower radius the a 

 axis. Base-lines corresponding to Ox of figures 13, 11 and 15 

 will be transposed to the horizontal diameter, as in figure 15, 

 and indicated by correspondence in lettering. The graphical, 

 solutions were all executed carefully with fine pencil lines and 



