96 Pear-shaped Configurations of Equilibrium [en. v 



and these equations, in combination with the first and second order equations 

 already discussed, express the condition that the third order figure (253) 

 shall be a configuration of equilibrium. 



97. The numerical discussion of these equations proves long and tedious*. 

 We first write down the values of C n , C 12 , etc. by a comparison of equations 

 (256) and (258). As a typical coefficient may be given the value of C i2 which 

 is found to be 



/3 + nABa 



- / AZ^ 



l|AC 2 a 7 2 + 1 JABCa/3 7 + 



+ 5f A 2 B ay 8 2 + l|A^Ca/3 7 + 2i|AB^ 



liABC/3 2 7 ] 



f AC (k + m/3 + n 7 ) + |BC (My 

 T ^C 2 (Na + 2ny) 



|AB (Z + 2na) + |AC (Ly + 2ma) -f JBC (ia + m/3 + ny)] d\ 



and on computing the numerical values of the various terms this reduces to 



Cl2 = _ 0-0002799H - 0-0093206 Jtt + 0-0103775N 



- 0-00458151 - 0'0016151m + 0'0040268n + 0'0042388. 



The remaining C-coefficients may be similarly evaluated, and equations 

 (260) then become a system of six linear "equations from which to determine 

 the six unknowns H, jftft, "N, I, m, n. The solution of these equations is 

 found to be 



H = - 12-6275, Jtt = - 0-0307056, N = - 0-0044636, 



I = - 0-0116194, m = 0-42602, n = 1-15365. 



* For fuller details than are given here see Phil. Trans. 217 A, p. 20. 



