196 ON THE DETERMINATION OF THE ATTRACTIONS 



Supposing therefore that U like V also satisfies the equation 

 (2'), each of the expressions (11) and (12) will be reduced to its 

 upper line, and we shall get by equating these two forms of the 

 same quantity : 



the quantities bearing an accent belonging, as was before ex- 

 plained, to one of the extreme limits. 



Because V satisfies the condition (9), the equation imme- 

 diately preceding may be written 



If now we give to the general function U the particular 

 value 



l-n 



which is admissible, since it satisfies for V to the equation (2), 

 and gives U" = 0, the last formula will become 



du 



dx,dx,...dx 8 .(l-n)u >n -"V 



n+i \ L{ >)> 



in which expression u' must be regarded as an evanescent posi- 

 tive quantity. 



In order now to effect the integrations indicated in the second 

 member of this equation, let us make 



05, 05/' = up cos 6 1 ; a? 2 a? 2 " = up sin O l cos 6 Z ; 



o? 3 o? 3 " = up sin 0j sin 2 cos 3 , &c. 

 until we arrive at the two last, viz., 



#,_! x" t _ l = up sin O l sin 2 . . . sin 0,_ 2 cos 0,_ 1} 

 x . & up sin 6 \ sin 2 . . . sin 0,_ 2 sin ^ 

 w' being, as before, a vanishing quantity. 



