280 



♦ KNOWLBDGE ♦ 



[October 1, 1887. 



have precisely the same construction for any point whatever, 

 either on or within the globe. The point n, for instance, 

 moves to n' and back to n, at a rate exactly proportional 

 to that at which P moves to p and back to P. 



Thirdly, every line through aOa' describes a double cone 

 about aOa'. The figure of this double cone for the line 

 POP' is shown ; and it is clear that not only would it be as 

 easy to show the cone for any other line through O, but also 

 for a line through any other point in aOa'. For instance, 

 the IhiP. cc', which crosses aOa' at a point near R', describes 

 a double cone, of which this point is the vertex, and of 

 which the upper is the lesser portion. The line a a' de- 

 scribes a simple cone of which a is the vertex. Finite lines 

 which irould have to he produced to meet aOa' produced, 

 describe only frusta of cones ; but, mathematically speak- 

 ing, we say that every line through any point of the line of 

 which aOa' is a part, describes a double cone around this 

 line as axis. 



Fourthly, bb', aa', cc', tt', etc., represent circles around 

 the axis POP', that is, they represent latitude-parallels. 



having this line for major axis, and touching the Hues e e 

 and e'e', the outline of the equator is determined to be 

 that shown in fig. 2. In this case E e'E', fig. 2, is the 

 nearer part of the ellipse, the lighter half of the ellipse 

 lying on the farther half of the globe, supposed for the 

 moment to be transparent : but if we had supposed the 

 rotation to have been through an angle PolO (not the acute 

 angle, but the angle measured by the arc PPiplO), we should 

 have obtained the same shape and position for our ellipse, 

 only the lighter part would have been on the nearer half, 

 and Ee'E' on the further half. 



The case of a latitude-parallel is just as easily examined. 

 Take the parallel c n c', for instance. When P is at r, n, the 

 centre of the parallel, is at n, and a line through n at right 

 angles to rr', equal in length to the line cc', and bisected at 

 n, is the major axis of the ellipse representing the latitude- 

 parallel cc'. Further, c will be the point farthest from, and 

 c' the point nearest to the plane tOt', throughout the rotation 

 we are considering ; and the ellipse will therefore always touch 

 the lines cc, c'c', figs. 1 and 2. If, therefore, in fig. 2 we 



Fig. 1. 



Let us examine into the apparent shape of the ellipses 

 into which these circles are projected as the rotation 

 we are considering goes on. Take first the equator, 

 represented by e O e'. The points e and e' are those 

 farthest from the plane 1 1', and it is obvious that they 

 will remain so throughout the rotation. Therefore if the 

 lines e e, e'e' are drawn parallel to t O t', the ellipse 

 representing the equator must always touch these two lines. 

 Further, since it is easy to find the positions assumed by the 

 points e and e' (by following the plan already described for 

 P) for a given amount of rotation, we immediately find the 

 points at which the ellipse touches the lines e e and e'e'. It 

 is also obvious that the greater axis of the ellipse cannot but 

 be at right angles to the polar axis of the globe, and we 

 have seen how the position of this axis is determined. 

 Suppose, for instance, that the rotation around aOa' has 

 taken place through an angle equal to the angle Po 2, then 

 the north pole has come to r, and the south pole to r', 

 therefore r r' is the position of the polar axis. If, then, 

 in fig. 2 we draw POP' in this position, the line EOE' at 

 right angles to POP' is the major axis of the ellipse repre- 

 senting the equator. And as it is easy to draw an ellipse 



Fig. 2. 



take n, determined by drawing mxn parallel to 00, to meet 

 PP', fig. 2, and take CC in fig. 2 equal to cc' in fig. 1, at 

 right angles to PP' and bisected in n, an ellipse, Cc'C, 

 having C'C as gi-eater axis and touching the lines c c, c'c', 

 represents the latitude-parallel required.* The part Cc'C is 

 the nearer, and it is clear that more than one-half of the 

 parallel lies on the hemisphere turned towai-ds the observer. 

 In this way every latitude-parallel is determined. The 

 parallel a a' presents a peculiarity worthy of notice. Since 

 this parallel has upon it a point, a, which is the ujjpermost 

 point of the globe, and since this point cannot but remain 

 uppermost throughout the motion, the ellipse representing 

 this parallel always appears to touch the outline of the 

 globe's disc in the point a. 



* It is easily seen that the points of contact of all the ellipses 

 with their corresponding pairs of parallels lie on the ellipse a t'a t 

 (fig. 2) ; since these points of contact, originally lying on the circle 

 to ae'a'e, must be brought by the rotation of this circle around aOa' 

 lie on an ellipse having aOa' as axis, and passing through the points 

 P and P'. Also it is obvious that the minor axis tOt' of the ellipse 

 a Ma t is easily determined by taking the angle a02, equal to the 

 angle Ro2 (lig. 1), and drawing 2 1' parallel to PP'. 



