By I.D. Faux, M.J. Pratt
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We shall do the latter. The advantage of this choice will be evident from what follows. We have still to remark, that if only the position of Sr and Si is determined, the position of S4 ensues from it not in one, but in two ways; for, a position of S4 gives no other positions of Sr and Si as its "opposite position" for which all vectors are reversed ; that opposite position can be obtained by an arbitrary equiangular double rotation over an angle π\ Sr and Si then rotate 2 π and are again in their former position.
2 n d . Three of the squares of inertia are equal and unequal to the fourth. We take the axis of the unequal one as X4 axis in S4. Then Ax = A2 = Az = A; ax = α, = α^ = a; bx = b2 = i 3 = />; and the equations (Λ) pass into b φ = — Vxp . φ a ψ: : — V ψ . ψ a therefore φ and ψ are both perpendicular to φ and to if', whilst φ-}-ψ=:0, so φ -|- ψ is constant and ) and ψ are each for itself constant in absolute value, so that they both rotate about their sum ("in space" that vector of the sum has in general quite a different position for Sr than for Si) by which the two "cones in the solid" are determined.
Proof. We suppose the limited space AS^+I to be provided with curvilinear coordinates ux... up+\ determined as intersection of curved Cp's, i. e. curved coordinatespaces of p-dimen^ions. We suppose the system of curvilinear coordinates to be inside the boundary without singularities and the boundary with respect to those coordinates to be everywhere convex. The integral element of P+XY becomes when expressed in differential quotients of PX : àœK : -yai V H l % Σ =ai ~VH dux ÔXa VH 9ft 'ρ+Ι du, du.