658 HENRY S. WASHINGTON 



As it is a somewhat new departure in petrography it may be of 

 use to others to outline the method which I have employed. 

 The curves shown in Diagram 3 were plotted on paper ruled in 

 inches and tenths. The particular scale is simply a matter of 

 convenience, and it is not necessary to reduce the percentages to 

 100, as we are dealing with relative amounts. 



The formula employed is well known, being the second of 

 Guldin's theorems, viz., the volume of the solid generated by 

 the revolution of a closed curve or plane figure about an axis in 

 its plane, but exterior to itself, is equal to the product of the 

 area of the generating curve into the path described by the cen- 

 ter of gravity of the revolving area. 



F= 2TTrA, 

 where F is the volume, rthe distance from the axis to the center 

 of gravity, and A the area of the plane figure. 



The areas of the curve, i. e., of the space embraced within the 

 curve itself, and the limits of the diagram, are easily found, 

 either by counting the squares, or by calculation of the area of 

 the trapezoids formed by the respective chords and the limiting 

 lines, and addition to these of the areas embraced between the 

 chords and the curves. 



The centers of gravity are found by dividing the trapezoids 

 into two triangles, and finciing their centers of gravity, when the 

 center of gravity of the trapezoid will be at the intersection of 

 the line connecting the centers of the two triangles and one con- 

 necting the middle points of the two parallel sides. In the 

 case of the more curved lines a correction must be made for the 

 area between the chord and the curve, but this will always be 

 small. SiOg was regarded as composed of the large rectangle 

 from .400 below the bottom of the diagram to .249, and the 

 space between this upper boundary and the inflexed curve. 



The resultant volumes, being based on the molecular ratios, 

 have to be multiplied by the molecular weights of the respective 

 oxides, in order to arrive at the percentage composition of the 

 whole. In this way I obtained the following figures, which are 

 pfiven in full to illustrate the method. 



