320 
PRO]'’rSSOP. Ct. H. DARWIX on the PEARt-SHAPED FTOURE 
H ence 
I [Wfin-) €/'(</>)]- i5c/or. 
Til order that the new figure may l)e one of eqnililjrium, this expression must oe 
stationary for variations of e. It follows that we must either have e = 0, which 
i/ 
leads hack to Jacobi’s ellipsoid, or else 
_ (V||) 
P]f l^o) Qf fPo) 
This last condition is what M. Poincare calls the vanishing of a coefficient of 
stability.^ It shows that if Pq and /3 satisfy not only the condition for the Jacobian 
ellipsoid, namely, (y,) == ('hi) Ql’ (Ej)^ also this equation, we have 
arrived at a figure which belongs at the same time to two series, and there is a 
bifurcation at this point. The form of the figure is found by attributing to e any 
arbitrary hut small value. 
8K= - 
1 — 
PJ (E)) QJ(''o)_ 
§ 6. 71ie Properties of the Successive Coefficients of Stabilitij. 
Corresponding to each harmonic deformation of the ellipsoid, there is a coefficient 
of stability of one of the two forms 
_ ^ _ Pj(ro)Qy (yj 
PJl'^olQiPzy) PJlJ'nlQiMz'n) ' 
These coefficients may be written it/ or Kf according to an easily intelligible 
notation. Tlie Jacobian ellipsoid is defined by yj, and the question arises as to the 
jiossibility of the vanishing of the several TVs as y, gradually diminishes from infinitv, 
that is to say, as the ellipsoid lengthens. 
An harmonic of the first order merely denotes a shift of the centre of inertia 
along one of the three axes; one of the second order denotes a change of ellijiticitv 
of the ellipsoid. Since we must kee}) the centre of inertia at the origin, and since 
the ellipticity is determined by the consideration that the ellijisoid is a Jacobian, 
these harmonics need not he considered, and we may begin with those of the 
third order. 
I shall not attempt to follov* M. Poincare in his masterly discussion of the pro¬ 
perties of the coefficients of stability,! hut will merely restate in my own notation 
the principal conclusions at which he has arrived. 
* ‘Acta Math.,’ vol. 7, 1885, p. .32E The factors -I and 1 '2ii + 1 (or 1 2i. + 1, if / is the degree of 
the harmonic) whicli occur in his form of the condition are included in mj" functions. 
t Sections 10 and 12 of his memoir. T Inu'e to thank him for saving me from making a serious mistake 
in this portion of my work. 
