98 Professor A. E. H. Love [March 6, 



a map of one hemisphere with its centre in the middle of France, and, 

 over this map, a map of the other hemisphere turned upside doiun, so 

 that any point of the second map coincides with its antipodes in the 

 first. [Shown by a sHde in which the outhnes of the two maps were 

 coloured differently.] The combined map shows oceans nearly every- 

 wdiere at the antipodes of continents. But it shows Borneo and some 

 adjacent islands at the antipodes of parts of South America ; it 

 shows the southern extremity of South America antipodal to a part 

 of Asia, and it shows the Antarctic region of elevation at the anti- 

 podes of the Arctic one. Thus certain parts of the continental 

 region of elevation are antipodal to certain others. Now this ar- 

 rangement assures us that the surface is somewhat ellipsoidal, or that 

 it has an inequality specified by a spherical harmonic of the second 

 degree. You know that an ellipsoid is a surface specified by means 

 of three principal directions, which I may refer to as right and left, 

 back and front, up and down. In one of these, say the right and 

 left direction, the ellipsoid projects beyond a sphere of the same 

 volume ; in the second, back and front, it runs inside the sphere ; 

 in the third, up and down, it may project beyond or run inside 

 according to circumstances. The right and left direction obviously 

 corresponds with the diameter from Borneo to the north-east corner 

 of Brazil ; the up and down with the earth's axis. The main 

 feature of a nearly spherical ellipsoid is the existence of two great 

 areas of depression, antipodal to each other, corresponding with the 

 parts where the eUipsoid runs inside the sphere. I make out that 

 the best fit is obtained by fixing one of these so as to coincide nearly 

 with the basin of the Pacific Ocean. The antipodal depression must 

 then contain Africa, the Mediterranean, some neighbouring countries, 

 and parts of the Atlantic and Indian Oceans. It appears that the 

 ellipsoidal inequality, though it certainly exists, is less important than 

 the inequality of the first degree. [Slides were shown of maps of 

 two hemispheres with the two areas of depression marked.] 



Now we go back to our double map and observe that Australia is 

 antipodal to the central part of the Atlantic Ocean. [Double map 

 with centre of one hemisphere in Australia shown by slide.] Near 

 the middle of the w^ater hemisphere we have a continent surrounded 

 by ocean. Near the middle of the land hemisphere we have an ocean 

 almost surrounded by continents. This arrangement cannot be 

 expressed by the harmonic of the first degree, or by that of the 

 second degree, or by any combination of the two. It shows to a 

 mathematical eye that there must be an inequality specified by a 

 spherical harmonic of the third degree. The simplest kind of defor- 

 mation of a sphere which can be expressed by a spherical harmonic of 

 the third degree is shown in Fig. 1, p. 99, where the dotted circle repre- 

 sents the sphere. [Shown by slide.] The figure has been described 

 as pear-shaped. Beginning at the top we see the elevated stalky the 

 depressed waist, the protuberant ring, and the flattened croivn. A 



