98 Pear-shaped Configurations of Equilibrium [CH. v 



while the value of S 3 may be at once written down by interchange of letters 



in 



The three equations (261) can now be written in the form 



e o r Adx , r d\ \ . , . r Bd\ . , r c^x \ 



J o A A 2 



00 G 



d\ 



<&, 



&AB 



r* 



= 8 2 -2 -^ 

 Jo AA 



> (263), 



rfx 



l\ \ 



icv 



-t-rnB* 



J t-^-n-v 



and p, q, r do not occur in the right-hand members of these equations. 



99. These equations appear at first sight to be a system of three simple 

 linear equations determining p, tj, t, but this proves not to be the case. Let 

 the equations be written for brevity in the form 



(264). 



Then, by simple transformation of the integrals, the values of the co- 

 efficients k it ki t - ar e found to be 



_/ 



l ~'*~ Cz ~ 



_3_ 



4a 2 



2(9 



46 2 ^' 



k " - ~ 



** ~ 2 



- -- - - 



(265), 



in which c lt c 2 , c s are the integrals already specified in equations (217). 



With these values assigned to k L , &/, ... it will be found that equations 

 (218) (220), which are the equations determining the third-harmonic dis- 

 placement at the point of bifurcation, assume the form 



= 

 = J 



.(266). 



