Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/41

Rh the $$\text{X-}, \text{Y-}$$, and $$\text{Z-}$$components of $$\lambda$$ and adding. Hence the second equation may be regarded as the most general form of a scalar equation of the first degree in $$a, b, c, d,$$ etc., which can be derived from the original vector equation or its equivalent three scalar equations. If we wish to have two of the scalars, as $$b$$ and $$c$$, disappear, we have only to choose for $$\lambda$$ a vector perpendicular to $$\beta$$ and $$\gamma$$. Such a vector is $$\beta \times \gamma$$. We thus obtain 37. Relations of four vectors.—By this method of elimination we may find the values of the coefficients $$a, b$$, and $$c$$ in the equation by which any vector $$\rho$$ is expressed in terms of three others. (See No. 10.) If we multiply directly by $$\beta \times \gamma, \gamma \times \alpha$$, and $$\alpha \times \beta$$, we obtain whence  By substitution of these values, we obtain the identical equation,  (Compare No. 31.) If we wish the four vectors to appear symmetrically in the equation we may write  If we wish to express $$\rho$$ as a sum of vectors having directions perpendicular to the planes of $$\alpha$$ and $$\beta,$$ of $$\beta$$ and $$\gamma$$, and of $$\gamma$$ and $$\alpha$$, we may write  To obtain the values of $$e, f, g$$, we multiply directly by $$a$$, by $$\beta$$, and by $$\gamma$$. This gives Substituting these values we obtain the identical equation  (Compare No. 32.)

38. Reciprocal systems of vectors.— The results of the preceding section may be more compactly expressed if we use the abbreviations The identical equations (4) and (8) of the preceding number thus become