The case of three degrees of freedom is instructive on account of the geometrical analogies. With a view to these we may write
| 2T = aẋ2 + bẏ2 + cż2 + 2fẏż + 2gżẋ + 2hẋẏ, |
| 2V = Ax2 + By2 + Cz2 + 2Fyz + 2Gzx + 2Hxy. |
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It is obvious that the ratio
| V (x, y, z) |
| T (x, y, z) |
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must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive. Denoting this value by σ12, we have
| Ax1 + Hy1 + Gz1 = σ12 (ax1 + hy1 + ∂gz1), |
| Hx1 + By1 + Fz1 = σ12 (hx1 + by1 + fz1), |
| Gx1 + Fy1 + Cz1 = σ12 (gx1 + fy1 + cz1), |
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