ANNIVERSARY ADDRESS OF THE PRESIDENT. XXXI 



determined by mathematical calculation founded on the above or 

 some equivalent reasoning ; but where less accuracy is required, the 

 process above indicated may be graphically performed on a common 

 globe, so as to determine the position of the pole with sufficient ex- 

 actness. 



It will further be necessary for us to consider one or two points 

 concerning the relations in which two such systems as I have described 

 stand to each other. Since the planes of their respective great circles 

 both pass through the centre of the sphere, they must intersect along 

 a diameter of the sphere, and the circles themselves must intersect 

 each other at the two extremities of that diameter. Either of these 

 points of intersection is one of the elements by which the position 

 of one of these great circles may be determined with respect to the 

 other. The second element is the angle at which they intersect. 

 If these two elements — one of the points of intersection, and the 

 angle of intersection — be known, the relative positions of the two 

 great circles, or of the two corresponding systems of small circles, are 

 completely determined. These relative positions may also be ex- 

 pressed by that of the poles of the two systems, but the above is 

 probably more easily conceived. 



The great circle representing the ecliptic on ordinary terrestrial 

 globes, and the small circles parallel to it (usually represented on 

 celestial globes), aiford a familiar example of a second system of 

 small circles. The points representing the equinoxes are the points 

 of intersection. It will also be observed that the ecliptic touches 

 the two small circles which represent the tropics, but vpithout inter- 

 secting them, and at equal distances (90°) from the equinoxes. It is 

 also easily seen that the directions of the great and the small circles 

 at their common point of contact will be identical ; in other words, 

 the straight line which is a tangent to the one at that point vdll also 

 be a tangent to the other. This fact may also be stated in another 

 way : — if we take a very small arc, a portion of the great circle, the 

 middle point of that arc being the point of contact, and a similar 

 equally small arc of the small circle, these two small arcs will be very 

 approximately coincident, and may be considered so for all practical 

 purposes vdth which we are here concerned. 



The propositions which have here been stated with reference to a 

 particular point of a particular small circle, are equally true for every 

 point of every small circle ; for a great circle can be drawn so as to 

 touch without intersecting any small circle belonging to any system 

 at any proposed point, and this point of contact will necessarily be 

 90° from either point of intersection of this great circle with that of 

 the system to which the small circle thus touched belongs. 



We may now distinctly explain the modified sense in which our 

 author uses the term parallelism as applied to mountain chains or 

 other lines of elevation situated on any part of the earth's surface. 

 Every such line is considered as a portion of a great circle, but is 

 in general so short that it may be regarded as coinciding with the 

 tangent to its middle point, or with a small circle having the same 

 tangent at that point, in the same manner as the great circle repre- 



