6 Trans. Acad. Sci. of St. Louis. 



solid spheroid of the earth and which we have heretofore 

 assumed to be in a state of equilibrium, being really in 

 motion. The direct influence here referred to gives rise to a 

 force <l>j which is the resultant of 1 ) the forces of attraction 

 of the particle by the sun and the moon and 2) the forces 

 equal but opposite in direction to the forces of attraction by 

 the sun and the moon of a mass equal to that of the given 

 particle and placed in the center of the earth. The indirect 

 influence of the sun and the moon through the medium of 

 the tides gives rise to another, also very small, force <t>g. 



Let <E> be the resultant of these forces 4)^ and ^^. It will, 

 of course, vary with the time, but for a comparatively small 

 interval of time, as in the case of experiments and observa- 

 tions on the surface of the earth we may assume the force 4>,. 

 as well as its projections ^j., 4> , ^^on the coordinate axes> to 



be constant.* A rough computation will enable the reader 

 to see that the forces ^^, 4> , <I>^ are small quantities of the 



order of gg . To obtain the exact values of the components of 



the solar and lunar attraction it is necessary to resort to the 

 rigorous methods of Celestial Mechanics. We will assume 

 that these components have been actually computed for a 

 given epoch Iq. Then these computed values, which we will 

 denote by ^^ , ^ , 4>^^, may be substituted for the forces 



<I>|,, <l> , <E>c. during the comparatively small interval of time 



under consideration. 



7. The differential equations which express that a con- 

 strained particle is in a state of equilibrium relatively to the 

 earth can now be written as follows : 



• In this we follow the example of Puiseux. See his paper entitled 

 " De l'4qailibre et du mouvement des corps p^sants en ayant 6gard aux 

 variations de direction et d'intensit^ de la p^santeur." Annales de I'ficole 

 Normale. 2« S6rie. 1. 1. 1872. p. 23. 



