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SCIENCE 



[N. S. Vol. XXVI. No. 669 



the center of gravity, there would be an 

 important secondary effect. The gravita- 

 tional attraction of an ellipsoid differs from 

 that of a sphere, and it may be represented 

 as the attraction of a sphere together with 

 an additional attraction. If the density 

 was greater in one hemi-ellipsoid than in 

 the other, the additional attraction would 

 produce a greater effect in the parts where 

 the density was in excess, and the result, 

 just as in the ease of rotation, would be a 

 furrowing of the surface. It has been 

 proved that the formula for this furrowing 

 also is expressed by a spherical harmonic 

 of the third degree. 



We are brought to the theory of spher- 

 ical harmonics and the spherical harmonic 

 analysis. Spherical harmonics are certain 

 quantities which vary in a regular fashion 

 over the stirface of a sphere, becoming 

 positive in some parts and negative in 

 others. I spoke just now of making a 

 model of a nearly spherical surface by 

 removing material from some parts and 

 heaping it up on others. Spherical har- 

 monics specify standard patterns of de- 

 formation of spheres. For instance, we 

 might remove material over one hemis- 

 phere down to the surface of an equal 

 but not concentric sphere (cf. Fig. 5) and 

 heap up the material over the other hemis- 

 phere. We should produce a sphere equal 

 to the original but in a new position. The 

 formula for the thickness of the material 

 removed or added is a spherical harmonic 

 of the first degree. It specifies the simplest 

 standard pattern of deformation. Again, 

 we might remove material from some parts 

 of our model and heap it up on other parts 

 so as to convert the sphere into an ellipsoid. 

 The formula for the thickness of that 

 which is removed or added is a spherical 

 harmonic of the second degree. Deforma- 

 tion of a sphere into an ellipsoid is the 

 second standard pattern of deformation. 



The mathematical method of determining 

 the appropriate series of standard patterns 

 is the theory of spherical harmonics. Its 

 importance arises from the result that any 

 pattern whatever can be reached by first 

 making the deformation according to the 

 first pattern, then going on to make the 

 deformation according to the second pat- 

 tern, and so on. If we begin with a pat- 

 tern, for instance the shape of the earth, 

 which is not a standard pattern, we can 

 find out how great a deformation of each 

 standard pattern must be made in order 

 to reproduce the prescribed pattern. The 

 method of doing this is the method of 

 spherical harmonic analysis. Except in 

 very simple cases the application of it in- 

 volves rather tedious computations. With 

 much kind assistance and encouragement 

 from Professor Turner, I made a rough 

 spherical harmonic analysis of the earth's 

 surface. I divided the surface into 2,592 

 small areas, rather smaller on the average 

 than Great Britain, gave them the value 

 -j- 1, or one unit of elevation, if they are 

 above the sea, and the value — 1, or one 

 unit of depression, if they are below the 

 1,400-fathom line. To the intermediate 

 areas I gave the value 0. The distribu- 

 tion of the numbers over the surface was 

 analyzed for spherical harmonics of the 

 first, second, and third degrees. 



Any spherical harmonic of the first de- 

 gree gives us a division of the surface into 

 two hemispheres— one elevated, the other 

 depressed. The spherical harmonic an- 

 alysis informs us as to the position of the 

 great circle which separates the two hemis- 

 pheres, and also as to the ratio of the 

 maximum elevation of this pattern to the 

 maximum elevation of any other pattern. 

 The central region of greatest elevation of 

 this pattern is found to be in the neigh- 

 borhood of the Crimea, and the region of 

 elevation contains the Arctic Ocean and 



