92 Dr. C. V. Burton on a 



and at any interior point distant r from the centre 



^(1-rA^) (2) 



a 



3. At any interior point, a mass-element free to move 

 under the periodic term of the field-intensity (2) would 

 execute oscillations given by 



Br = -■£*&* (3) 



Consider then the radial vibrations defined by (3), and the 

 forces which would be needed to maintain them. At any 

 given instant, every radius vector r has acquired an incre- 

 ment hr proportional to r, the whole sphere being thus 

 symmetrically and homogeneously expanded or contracted, 

 and the proportional increment of linear dimensions being 

 —gjap 2 . Ae ipt , corresponding to the proportional increment 

 of volume 



Sy/ap 2 . Ae** (4) 



In the sequel the discussion is restricted to the case where 

 Ap 2 is small enough to make the expression (4) a small 

 fraction. 



4. Thus if every mass-element suffers the periodic changes 

 of position defined by (3) , there will also be for each element 

 periodic (proportional) changes of volume indicated by (4). 

 The changes of position we may consider to be provided for 

 by the periodic term in (2), while throughout the mass of the 

 planet, the fluctuations of pressure needed to produce the 

 changes of volume of the various elements are supplied by 

 the reaction of those elements on one another. At the free 

 surface, however, we should further require a periodic pres- 

 sure P given by 



V^UgjapKAe^ (5) 



to complete the system of forces under which the vibrations 

 (3) would be executed ; k being the bulk-modulus of elasticity 

 of the substance of the planet. 



5. For a moment suppose the periodic pressure P to be 

 applied to the earth's surface, in addition to the partly periodic 

 character of gravitation expressed by the factor l-\-Ae ipt . 

 Then every mass-element of the earth would be executing a 

 vibration expressed by (3), precisely as if it were free to 

 move under the periodic part of the earth's attraction, and a 

 relatively small body resting on the earth's surface would 

 execute, under gravitative influence alone, vibrations exactly 



