292 Prbf. Thomson on the Muttial Attraction or Repulsion 

 ing manner. We have by (c), (b), and (a), 



P,Q,+P9Q»-.+P3Q»-.+&c.=iQ„+('-=g=^V,+l)Q„_, 



( 



c^-a'-b'' 



-Pi)q«- 



,+ &c. 



= ^ 8,-1+ '^ "2 ^V tQ.-. + PsQ.-.+&c.) -{P,Q«-2+P»Q«-3 +;&c.); 

 and similarly we find 



^ Q^ q2 ___ ^2 



= -T S„_, H -T (SiSrt_2 + SgSn-a + &C.) — (SiS„_3 + S2S„.4 + &C.) j 



S,P»-i + S^Pn-a 4- SgPn-s + &C. 



= ^P..,+ ''~f~^' (S^P.., + S,P„,3 + &c.)-(StP»-3 + SaPn-4 + &c.); 



and 



SjQft- 1 + S2Q„-2 + SgQ;i_3 + &c. 



= 7rQn-i + 



ah 



ah 



(S,Q„_2 + S2Q„_3 + &c.)-(SiQ«-3+S2Q„-4 + &c.). 



Hence, if we put 



2;::(p.-A)+2;:r'(s„-<s,)=3s; 



,<=»» 



and 



s;;r(p.-«s,)=p'„ 



2;:r'(Q„.<s,)=Q„ 



• w. 



in terms of which notation the expression (15) for F becomes 



(S), 



we have 



- ah 



P' ,= 



C2_fl2_^2 



F - ir -— P ^ 



^ • • (3)- 



