Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/95

Rh when $$\text{N},$$ which is a whole number, is increased indefinitely. That this definition is equivalent to the preceding, will appear if the expression is expanded by the binomial theorem, which is evidently applicable in a case of this kind.

These functions of $$\Phi$$ are homologous with $$\Phi.$$

172. We may define the logarithm as the function which is the inverse of the exponential, so that the equations are equivalent, leaving it undetermined for the present whether every dyadic has a logarithm, and whether a dyadic can have more than one.

173. It follows at once from the second definition of the exponential function that, if $$\Phi$$ and $$\Psi$$ are homologous, and that, if $$\text{T}$$ is a positive or negative whole number,  174. If $$\Xi$$ and $$\Phi$$ are homologous dyadics, and such that the definitions of No. 171 give immediately  whence  175. If $$\Phi. \Psi = \Psi. \Phi = 0,$$ Therefore  176. For the first member of this equation is the limit of If we set $$\Phi = \alpha i + \beta j + \gamma k,$$ the limit becomes that of  the limit of which is the second member of the equation to be proved.

177. By the definition of exponentials, the expression represents the limit of