and Mathematics to Seismology. 191 



tion ; but in an elastic earth allowance must be made for the 

 fact that the attraction of the disturbed earth is not along the 

 normal. 



The effect on astronomical observations is still more com- 

 plicated. Thus let an observer take the altitude of a star in 

 the same vertical plane as the moon, using a mercury surface 

 for his horizontal plane. The observed altitude will differ 

 from the theoretical — i. e. the true altitude if the disturbing 

 influence were absent — by an amount equal to the angle 

 between the disturbed and undisturbed mercury surfaces. 

 This is the algebraical sum of the inclination of the resultant 

 gravitational force to the radius-vector in the disturbed con- 

 dition and of the inclination of this radius-vector to its undis- 

 turbed position. 



This explanation will show what the quantities are of 

 which we require to know the theoretical values. 



§ 24. To return to the problem. The earth is treated as 

 truly spherical when undisturbed, u centrifugal force " being 

 neglected, and as posssesed when disturbed of uniform den- 

 sity p, and of uniform isotropic elastic qualities throughout, 

 determined by the elastic constants m, ?i. 



The assumption of natural sphericity and the neglect of the 

 centrifugal force answer merely to the neglect of small quan- 

 tities of the second order of magnitude relative to those of the 

 first ; the other assumptions have been discussed in § 2. In 

 our ultimate applications the material will be supposed incom- 

 pressible, i. e. n/m = 0, but it is undesirable to introduce 

 unnecessary limitations in the mathematical results themselves, 

 Further, absolutely incompressible material is merely a ma- 

 thematical fiction, so it is desirable to have the means ready 

 to hand to apply a correction to mathematical results based 

 on such an hypothesis. 



Supposing the typical term in the potential of the disturbing- 

 forces to be 



r'V/ai, (43) 



where g { is a known surface-harmonic of degree i, and V/ a 

 given numerical magnitude, we easily see that the equation 

 to the strained surface will take the form 



r=a + ^ai<Ti. (44) 



A t the present stage all we know is that a t is small compared 

 to a, the mean radius of the strained surface. 



The bodily forces consist in part of the disturbing forces, 

 but mainly of the self-gravitational action of the " earth." 



