330 
PROFESSOR G. H. DARWIN ON THE PEAR-SHAPED FIGURE 
M. SoHWARZSCHiLD has remarked* that it is not absolutely certain that the 
principle of exchange of stability holds with reference to this figure, and that we 
cannot feel absolutely certain that the pear is stable unless we can prove that the 
moment of momentum is greater than in the critical Jacobian. 
With reference to this objection, M. Poixcare writes to me as follows :— 
“ Faisons croitre le moment de rotation, que j’aj^pellerai M, Deux hypotheses sont 
possibles. 
“ Ou bien pour M < (the moment of momentum of the Jacobian), nous aurons 
une seule figure, stable, a savoir rellq3soide de Jacobi, et pour M > Mg trois figures, 
une instable, fellipsoide, et deux stables (d’ailleurs egales entre elles), les deux figures 
pyriformes. 
“ Ou bien pour M < Mg, nous aurons trois figures d’equilibre, deux pyriformes 
instables, une stable, Fellipsoide, et pour M < Mg une seule figure instable, Fellqisoide— 
auquel cas la masse fluide devrait se dissoudre par un cataclysme subit. 
“ II y a done a verifier si pour les figures pyriformes, M > ou < Mg.” 
It seems very improbable that the latter can be the case; but this opinion is not a 
proof 
Since is stationary for the initial pear, a small change in the angular velocity 
will certainly produce a great change in the figure of the pear. If this investigation 
has, in fact, its counterpart in the genesis of satellites and planets, it seems clear that 
the birth of a new body, although not cataclysmal, is rapid. 
§ 9. Summary. 
It is possible by the methods explained in my previous paper on “ Harmonics ” to 
form rigorous expressions for the ellipsoidal harmonics of the third degree. Accordingly 
in § 1 I proceed to form those functions. In § 2 the notation is changed with a view 
to convenience in subsequent work, and for the sake of completeness the harmonics 
of the first and second degrees are also given. In § 3 the corresponding solid 
harmonics are expressed in rectangular co-ordinates x, y, z. In § 4 I find the 
Q-functions, the harmonic functions of the second kind, and express the results in 
terms of the elliptic integrals E and F. It appears that both the P- and Q-lunctions 
of the third degree of harmonics occur in three pairs which have the same algebraic 
forms, and that in each pair one of them only differs from the other in the value of a 
certain parameter. There is, lastly, a seventh function which stands by itself; this 
last corresponds to the solid harmonic xyz. 
In § 5 the equations for Jacobi’s ellipsoid are determined by the consideration that 
the energy must he stationary, and the superficial value of gravity is found in terms 
of the appropriate P- and Q-functions. I then proceed to find the additional terms 
* “ Die PoincaiAsche Theorie des Gleiehgewichts,” ‘ Aiinalen der K. Sternwarte, Miinchen,’ Bd. HI. 
