ON THE INTEGRATION 

 OF CERTAIN DIFFERENTIAL EQUATIONS*. 



No. I. 



IT is shown in the theory of the earth's figure, that if the 

 pressure and density at any point be connected by the equation 



dp = kp dp, 



where Jc is a constant, then the ellipticity of the surface may be 

 deduced from the solution of the equation 



This equation is not easily integrated. La Place, in the 

 eleventh book of the Mecanique Celeste (v. 51), gives a solution 

 of it, but without demonstration ; and the lacuna thus left is not 

 supplied in the works on the subject generally made use of in 

 Cambridge. 



Mr Gaskin has however effected the integration of 



is integral, in finite terms (vide Hymers' Diff. JEq. p. 53), 

 and the proposed equation is a case of this one. But perhaps a 

 more direct analysis is preferable, as it enables us to extend our 

 method to two or three classes of equations of all orders. One 

 of these will be considered in the present paper another, the 

 solution of which admits of a remarkable symbolical form, will 

 be given in the next number of the Journal. 



We shall begin with the particular equation which occurs in 

 the theory of the earth's figure, both because from its physical 



* Cambridge Mathematical Journal, No. X. Vol. n. p. 169. November, 1840. 



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