ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 41^ 



Having thus proved that every function of the form (20) which 

 satisfies (19) will likewise satisfy (18), it will be more simple to deter- 

 mine the remaining coefficients of (j> from those of cp^^^ by means of 

 tlie last equation, than to employ the formula (21) for that purpose. 



Making therefore h infinite in (18), and writing ~ in the place 

 of K, we get 



where (22) comprises the — ^ — —!■ combinations which can be formed of 



1.2 



the s indices taken in pairs. 



If now we substitute the value of before given (20'), and recol- 

 lect that for the terms of the highest degree we have 2»«r = 7, we shall 

 readily get 



0=(7-2»«,)(7+2»?r+»-l)^™,,»,,....,+(7».+2)(»w,+l)^„^, „^+2,...„^...(22), 



from which all the remaining coefficients in will readily be deduced, 

 when those of the part 0'^' are known. 



10. It now remains, as was before observed, to integrate the ordi- 

 nary differential equation (17) No. 8. But, by the known theory of 

 linear equations, the integration of (17) will always become more simple 

 when we have a particular value satisfying it, and fortunately in the 

 present case such a value may always be obtained from by simply 



changing f, into ' , . In fact if we represent the value thus ob- 



tained by Ho we shall have 



cih ^' </e/«v(2«:')' 



and by a second differentiation 



3H2 



