458 MOSSOTTI ON THE FORCES WHICH REGULATE 



(1) 



1 (1) 1 dQ^ 



^1 r ' t dr 



the last of these quantities will satisfy the equation (4), and will be its 

 complete integral. 



If the successive substitutions are performed, and, for brevity's sake, 

 we make 



/ i — \\ ('■-') 



(0 (-ly ('-')_r ir' V^+mj^o-i) .. 



" " -L"TJ ^ r 1 -I' + i • 



«•• - \:%^i- "«• ~ 



we 



(o) p i _ 1-|.- 



which gives, in the particular case of « = t, a- — I 1 -| : — | , 



shall have 



(o) (1) (0 (2) (0 ^.^ 0) 



a. (.) a. dQ. a. d^Q. ^') d'Qi 



Now if we make 



(o) (1) (2) («■) 



C Of Uj a a. /\ ar' 



"i(^')=|-;;r + ;;73i« + ;:ri^«' +7^"/^ ' 



(«) (1) (2) ('■) 



where a is put instead of V/ —f—y we shall have 

 Q'.= r.O.(r')+F',ti'(r'), 

 and the expression for F may take the form 



(5) i^ = S"j^|-J^y^^^s"fl,(.') r>'''+^Zr' 



^ ^r, / *2 — j^ \ P„ sin 6' d^'d^< 



*./ r o r'" J 



* The brackets are here employed in the same way as in A'^andermonde's no- 

 tation. 



