142 THE ROYAL SOCIETY OF CANADA 



board is mounted on a frame which may be inclined to any position, 

 and by means of a beam of Hght a projection is made on to a rigid 

 glass plane where paper may be placed and a tracing of the shadow 

 readily obtained. The line of projection is at right angles to the 

 glass plane which may conveniently be called the plane of reference. 

 This plane is parallel to the initial position of the frame, or initial 

 plane, when the two defining angles of inclination (6 and 0) between 

 the frame and the plane of reference, each become equal to zero. 

 (See §4.) 



In such an arrangement it will be seen that these defining angles 

 of projection do not each have the same degree of freedom in the ad- 

 justment, although all possible inclinations of the plane as a whole, 

 may be produced. The plane in which lies some graph plotted with 

 rectangular axes x and y, may be pivoted about the origin into any 

 position, but in the cases discussed here, the arrangement is such that 

 all the possible positions of the x axis lie in a plane at right angles to 

 the initial plane (and obviously also to the plane of reference). For 

 this reason the projection, x' , of the x axis always lies along its initial 

 direction, while the projected y axis may make any angle with x' , 

 depending upon the position of the inclined plane. 



Let = the angle between the x axis and its projection, the Xi 

 axis. 

 = the angle between the y axis and y' , the initial position 



of the y axis before rotation. 

 0' = the angle between the y axis and its projection on the 



initial plane. 

 € = the angle between y' and the projection of y on the 

 initial plane. 



Observe that 6 and are the angles recorded on the apparatus. 



Let us determine the equation of the projection of the graph of 

 ^—f iy) with reference to the axes x' and y'. This is equivalent to 

 transforming the co-ordinates for a rotation of axes, and then making 

 the z co-ordinates which arise, equal to zero. It can easily be shown 

 that x = lix' and y^lox' + m^y' where li = cos (9, l2 = sin ^sin0, andm2 

 = cos 0; thus curves of the type y = f(x) and x = f(y) become m2y' = 

 f(lix)— I2X' and Iix' = f(l2x' + m2y') respectively. 



It is often convenient to take as axes of reference, y", the pro- 

 jection of y, and x" at right angles to it {y") in the initial plane and 

 thus making an angle equal to e with x . If this is done the trans- 

 formation is made by using 



x = Lix" + Miy", and y = M2y" 

 where Li = li cos e, Mi = — h, sin e, and M2 = m2 cos « — 12 sin ^ 



