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STUDIES FOR STUDENTS 



the major and minor axes of the ellipses compared with those of 

 the diameters of equivalent circles, the ratios between the last two, 

 and the ratio between the diameters of the original circles and the 

 minor axes of the equivalent ellipses, for rotations of a vertical 

 line at intervals of 5°.' This table is as follows, the positions of 

 the major axes of the ellipses being calculated to the nearest i ' : 



The facts of this table are graphically represented by Fig. 3 

 for a rotation of a vertical line amounting to 45°. 



It is of interest to note that the table shows that with the 

 least possible shearing the greater diameter of an ellipse is 

 inclined 45° to the vertical, or 45° is the smallest or limiting 



^ The position of the major axis of a flattened ellipse with reference to the position^ 

 of a vertical line rotated a definite amount, and vice versa, resulting from shearing,, 

 may be calculated from the following relation : The tangent of twice the angle 

 between a horizontal line and a rotated vertical line (complements of the angles of 

 column l) is equal to twice the tangent of the angle between a horizontal line and the 

 corresponding major diameter of the flattened ellipse (angles of column 3). Assigning, 

 any possible value to either angle, the other is easily calculable. 



