Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/218

202 If we set and  the equation reduces to  where $${a, b, c, e, f, g}$$ are constant, and $$\phi$$ a quadratic function of $$\text{L, M, N,}$$ for a given medium and light of a given period.

13. Now this equation, which expresses a relation between the constants of the equations of wave-motion (1), will apply, with those equations, not only to such vibrations as actually take place, but also to such as we may imagine to take place under the influence of constraints determining the type of vibration. The free or unconstrained vibrations, with which alone we are concerned, are characterized by this, that infinitesimal variations (by constraint) of the type of vibration, that is, of the ratios of the quantities $${\alpha_{1}, \beta_{1}, \gamma_{1}, \alpha_{2}, \beta_{2}, \gamma_{2},}$$ will not affect the period by any quantity of the same order of magnitude. These variations must however be consistent with equations (4), which require that Hence, to obtain the conditions which characterize free vibration, we may differentiate equation (16) with respect to $${\alpha_{1}, \beta_{1}, \gamma_{1}, \alpha_{2}, \beta_{2}, \gamma_{2},}$$ regarding all other letters as constant, and give to $${d\alpha_{1}, d\beta_{1}, d\gamma_{1}, d\alpha_{2}, d\beta_{2}, d\gamma_{2},}$$ such values as are consistent with equations (17). Now $${d\alpha_{1}, d\beta_{1}, d\gamma_{1},}$$ are independent of $${d\alpha_{2}, d\beta_{2}, d\gamma_{2},}$$ and for either three variations, values proportional either to $${\alpha_{1}, \beta_{1}, \gamma_{1},}$$ or to $${\alpha_{2}, \beta_{2}, \gamma_{2},}$$ are possible. If, then, we differentiate equation (16) with respect to $${\alpha_{1}, \beta_{1}, \gamma_{1},}$$ and substitute first $${\alpha_{1}, \beta_{1}, \gamma_{1},}$$ and then $${\alpha_{2}, \beta_{2}, \gamma_{2},}$$ for $${d\alpha_{1}, d\beta_{1}, d\gamma_{1},}$$ and also differentiate with respect to $${\alpha_{2}, \beta_{2}, \gamma_{2},}$$ with similar substitutions, we shall obtain all the independent equations which this principle will yield. If we differentiate with respect to $${\alpha_{1}, \beta_{1}, \gamma_{1},}$$ and write $${\alpha_{1}, \beta_{1}, \gamma_{1},}$$ for $${d\alpha_{1}, d\beta_{1}, d\gamma_{1},}$$ we obtain