Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/97

Rh Now, by No. 174, Since  Therefore  and  In like manner we find  180. If $$\alpha, \beta, \gamma$$ and $$\alpha ', \beta ', \gamma '$$ are reciprocals, and and $$\text{N}$$ is any whole number,  Therefore,  If $$a$$ and $$b$$ are unequal, and $$c$$ other than zero, we may add  181. If $$\alpha, \beta, \gamma,$$ and $$\alpha ', \beta ', \gamma '$$ are reciprocals, and and $$\text{N}$$ is a whole number,  Therefore  Unless $$b = 0,$$ we may add  182. If we suppose any dyadic $$\Phi$$ to vary, but with the limitation that all its values are homologous, we may obtain from the definitions of No. 171