94 Mr DAVIES on the Nature of the Hour-Lines 



1. When the latitude of the point of contact is small. Let BC be 

 the tangent of the latitude of the point of contact ; C D, at right angles to 



it, the radius of the generating sphere ; and 

 AEZ any value of L as before. Draw DP' 

 at right angles to DB. In this case, the in- 

 tersection of B C and D P' may be considered 

 as inaccessible, and the description of the 

 circles DN and CHP as practically impos- 

 sible. However, we are in possession of nu- 

 merous processes for drawing lines which shall tend to the inaccessible 

 intersection of two given lines; and by attending to the construction 

 which was employed for facilitating the demonstration of the general me- 

 thod, we shall see at once that the circles are not essential to the con- 

 struction of the problem itself, though, when they can be employed, they 

 abbreviate the operation considerably. The substituted process, then, may 

 be as follows. 



Draw CH perpendicular to ZH (ZH tending to P); CM parallel 

 to ZH, and equal to CD; and having joined MH, prolong CH till 

 H N = H M. Draw NR' tending to P, and make K/NX = BE 9. The 

 intersection of HZ, NX gives X, a point the curve. 



2. When the latitude is 0, or the plane touches the equator of the 

 hectemorial sphere. The same course of reasoning leads to a very simple 



construction of this case also. For the point 

 P having now become infinitely distant, the 

 lines DP', BC, ZH, NR' are all parallel. 

 The point H, moreover, coincides with Z, 

 and C with B ; and hence the point M falls 

 in BC, and N in BZ. Hence, 



Take M B = radius of generating .sphere, 

 and find BE 9 as before. Take Z, the point corresponding to A E Z, and 

 join M Z. Make ZN = ZM, and draw NR' at right angles to AZ; 



