1896.] 



On a Type of Spherical Harmonics. 



189 



Hence, the effect of elastic distortion is to prolong the period in the 

 ratio l + e'je : 1. Taking a spheroid of the same size and mean density 

 as the Earth rotating in a sidereal day, we find that the period is 

 extended from 232 days to 335 days if the rigidity be that of steel. 

 Assuming that in the case of the actual Earth the effects of hetero- 

 geneity are to still further prolong the period in the ratio 305 : 232, 

 we find for the period of oscillation of the Earth, supposed of the 

 rigidity of steel, 440 days. 



Now, observation* indicates that the earth is undergoing such an 

 oscillation as we have been discussing in a period of 427 days instead 

 of the period of 305 days indicated by theory on the assumption that 

 the motion would take place sensibly in the same manner as if the 

 earth were rigid. We see now that the divergence between theory 

 and observation can be explained by taking into account the elastic 

 distortions of the solid parts of the earth, and that the degree of 

 rigidity required to account for the observed period is slightly in 

 excess of the rigidity of steel. 



This explanation" has been previously offered by Professor New- 

 comb,')' who has treated the problem by a simple geometrical method. 

 The main object in undertaking the present analytical investigation 

 was to examine the validity of certain hypotheses made by Newcomb. 

 An examination of the type of oscillation indicated by our analysis 

 shows that the procedure adopted by Newcomb is legitimate, but 

 that there is a slight error in his assumed law of displacement of the pole 

 of figure due to centrifugal force. Our method has the additional 

 advantage of showing the degree of accuracy to which the results 

 may be expected to hold good. 



II. te On a Type of Spherical Harmonics of -unrestricted 



Degree, Order, and Argument." By E. W. Hobson, Sc.D., 

 F.R.S. Received December 23, 1895. 



(Abstract.) 



The type of harmonics considered in this memoir is 



m0 . U n m (fl), 



where u n ' n M satisfies the differential equation 



0, 



* Vide Chandler, ' Astron. Journal,' vols. 11, et seq. 



f « Monthly Notices of the Koyal Astron. Soc.,' March, 1892. 



