1908] on the Figure and Constitution of the Earth. 97 



are only nearly spherical, or nearly concentric, the surface of the 

 ocean must have inequalities of the nature of a heaping up of the 

 waters over areas of oceanic dimensions. The inequalities of the sur- 

 faces of equal density determine those of the surface of the ocean, 

 and have a decided influence upon those of the surface of the earth. 



In the simplest imaginable case the surfaces of equal density Avould 

 be accurately spherical but not accurately concentric, crowded together 

 on one side, spaced out on the opposite side. This arrangement may 

 be illustrated by a diagram of a system of circles one inside another, 

 with their centres in a straight line. [Shown by a slide.] The surface 

 of water resting on a body with such a distribution of mass would be 

 a sphere with its centre at, or very near to, the centre of gravity of 

 the body. [Shown by a slide.] It would cut the surface of the body 

 so as to yield a land hemisphere and a water hemisphere. A map 

 of one hemisphere of the earth, with its centre about the middle of 

 France contains all the continents except the southern part of South 

 America, the Antarctic continent, Australia ; the other hemisphere 

 is nearly covered by water. [Stereographic maps of these two hemi- 

 spheres were shown by slides.] It is certain that the distribution of 

 mass within the earth has an inequality of the type considered — the 

 type characterised by eccentric position of the centre of gravity. 

 This is the simplest kind of inequality which a nearly spherical and 

 symmetrical body can have. There is a mathematical theory by which 

 we can connect the inequalities of the surface with the distribution of 

 density. The standard patterns of inequalities are called spherical 

 harmonics. The kind of inequality which we have been considering 

 is specified by a spherical harmonic of the^rs^ degree. It is as if the 

 earth were drawn up out of the sea towards one side, the effect being 

 produced by gravity acting unsymmetrically, and drawing the sea to 

 one side of the earth. 



A\\ inequ^jlity of density specified by a spherical harmonic of the 

 first degree, means that the surfaces of equal density are crowded to 

 one side without change of shape. Inequalities of density specified 

 by spherical harmonics of higher degrees, mean that the surfaces of 

 equal density are distorted according to one or more of the standard 

 patterns. If this were the case in the earth, the surface of the earth, 

 and the surface of the sea, would be distorted according to the same 

 pattern, but the amounts of distortion would be different for the two 

 surfaces. This was exemplified in the case of the body with non- 

 concentric spherical surfaces of equal density, where the distortion 

 is replaced by a shifting to one side, and the surface of the body was 

 shifted to one side, the surface of the ocean to the other side. The 

 defects of the arrangement of land and water on the earth, as a land 

 hemisphere and a water hemisphere, enable us to detect other in- 

 equalities of the surface specified by spherical harmonics of higher 

 degrees. The defects are best shown by means of a map of the two 

 hemispheres drawn one on the top of the other. Thus, we may draw 



YoL, XIX, (No. 102) ^ II 



