(15)
c11α1α1′ + c22α2α2′ + ... + c12 (α1α2′ + α2α1′) + ... = 0,
(16)
provided the symbols αr, αr′ correspond to two distinct roots σ2, σ′2 of (6). To prove these relations, we replace the symbols A1, A2, ... An in (5) by α1, α2, ... αn respectively, multiply the resulting equations by a′1, a′2, ... a′n, in order, and add. The result, owing to its symmetry, must still hold if we interchange accented and unaccented Greek letters, and by comparison we deduce (15) and (16), provided σ2 and σ′2 are unequal. The actual determination of C, C′, C″, ... and ε, ε′, ε″, ... in terms of the initial conditions is as follows. If we write
C cos ε = H, −C sin ε = K,
(17)
we must have
| αrH + αr′H′ + αr″H″ + ... | = [qr]0, |
| σαrH + σ′αr′H′ + σ″αr″H″ + ... | = [q̇r]0, |
(18)
where the zero suffix indicates initial values. These equations can be at once solved for H, H′, H″, ... and K, K′, K″, ... by means of the orthogonal relations (15).