Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/37

Rh Again, if $$\alpha$$ be directed toward the east, and $$\beta$$ lie in the same horizontal plane and on the north side of $$\alpha$$, $$\alpha \times \beta$$ will be directed upward.

15. It is evident from the preceding definitions that

16. Moreover,$$[n\alpha]. \beta = \alpha. [n\beta],$$

and$$[n\alpha] \times \beta = \alpha \times [n\beta].$$

The brackets may therefore be omitted in such expressions.

17. From the definitions of No. 11 it appears that

18. If we resolve $$\beta$$ into two components $$\beta '$$ and $$\beta ''$$, of which the first is parallel and the second perpendicular to $$\alpha$$, we shall have 19. $$\alpha. [\beta + \gamma] = \alpha. \beta + \alpha. \gamma$$ and  $$\alpha \times [\beta + \gamma] = \alpha \times \beta + \alpha \times \gamma .$$

To prove this, let $$\sigma = \beta + \gamma$$, and resolve each of the vectors $$\alpha, \beta, \sigma$$ into two components, one parallel and the other perpendicular to $$\alpha$$. Let these be $$\beta ', \beta , \gamma ', \gamma , \sigma ', \sigma ''.$$ Then the equations to be proved will reduce by the last section to Now since $$\sigma = \beta + \gamma$$ we may form a triangle in space, the sides of which shall be $$\beta, \gamma$$, and $$\sigma .$$ Projecting this on a plane perpendicular to $$\alpha$$, we obtain a triangle having the sides $$\beta , \gamma ,$$ and $$\sigma ,$$ which affords the relation $$\sigma  = \beta  + \gamma .$$ If we pass planes perpendicular to a through the vertices of the first triangle, they will give on a line parallel to a segments equal to $$\beta ', \gamma ', \sigma '.$$ Thus we obtain the relation $$\sigma ' = \beta ' + \gamma '.$$ Therefore $$\alpha. \sigma ' = \alpha. \beta ' + \alpha. \gamma ',$$ since all the cosines involved in these products are equal to unity. Moreover, if $$\alpha$$ is a unit vector, we shall evidently have $$\alpha \times \sigma  = \alpha \times \beta  + \alpha \times \gamma '',$$ since the effect of the skew multiplication by a upon vectors in a plane perpendicular to a is simply to rotate them all 90° in that plane. But any case may be reduced to this by dividing both sides of the equation to be proved by the magnitude of $$\alpha .$$ The propositions are therefore proved.

20. Hence,