182 Prof, a H. Darwin. Stability of the Pear-shaped [June 19, 



The moment of inertia of the pear presents but little difficulty, since 

 it only involves those harmonic inequalities of J which are expressible 

 by harmonics of the second degree. On multiplying the moment of 

 inertia by ASm 2 we obtain the last contribution to the expression for E. 



The portion of E independent of <W 2 cannot involve e 2 , since the 

 vanishing of the coefficient of that term is the condition whence the 

 critical Jacobian ellipsoid was determined. If / denotes the coefficient 

 of any harmonic inequality other than the third zonal one, this portion 

 of E is found to contain terms in e?f, and (/) 2 . The coefficient of 

 ceo- consists of a constant term and terms in e' 2 , / 2 , f-f, where these f's 

 denote the coefficients of the second zonal and sectorial harmonics. 

 If / refers to any harmonic of odd degree, the coefficient of the 

 corresponding term in e 2 f vanishes. If, then, we make E stationary 

 for variations of the coefficient of any odd harmonic, that coefficient is 

 seen to vanish. Hence it follows that the expression for the pear 

 cannot involve any odd harmonic other than the third zonal one. 

 Conditions of symmetry also negative the existence of even harmonics 

 of the sine type, and of even harmonics of the cosine type but of odd 

 rank. 



On equating to zero the variations of E for all the remaining f's, 

 excepting f 2 and f 2 2 , we at once obtain their values in terms of e 2 . 

 Equating to zero the variations for e 2 , / 2 , f-2 2 , we obtain three equations, 

 which give Sto 2 , f-2, f-2 2 as multiples of e 2 . 



It seems unnecessary to explain here the methods adopted for 

 reducing the analytical results to numbers ; it may suffice to say that 

 the task was very laborious. 



The harmonic terms included in the computation were those of 

 degree 2, and ranks 0, 2 ; of degree 4, and ranks 0, 2, 4 : and of 

 degree 6, and ranks 0, 2, 4. The sixth sectorial harmonic would cer- 

 tainly have proved negligible. 



The expression for Sw' 2 was found in the form of a fraction, of which 

 the denominator is determinate, and the numerator is the sum of an 

 infinite series. Nine terms of this series were computed, namely, a 

 constant term and the contributions of the eight harmonics above 

 enumerated. The result shows that the square of the angular velocity 

 of the pear is less than that of the Jacobian in about the proportion 

 1 -le 2 to 1. 



On the other hand, the angular momentum is greater in about the 

 proportion of 1 + T V 2 to 1. If this last result were based on a 

 rigorous summation of the infinite series, it would absolutely prove the 

 stability of the pear. The inclusion of the uncomputed residue of 

 the series would undoubtedly tend in the direction of reducing the 

 coefficient given above in round numbers as — ^, and if it were to reduce 

 it to a negative quantity we should conclude that the pear is unstable 

 after all. 



