Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/63

Rh where $$r^2 dq$$ is the element of a spherical surface having center at $$\rho$$ and radius $$r.$$ We may also set We thus obtain  where $$\bar u$$ denotes the average value of $$u$$ in a spherical surface of radius $$r$$ about the point $$\rho$$ as center.

Now if $$\text{Pot } u$$ has in general a definite value, we must have $$\bar u' = 0$$ for $$r = \infty .$$ Also, $$\nabla. \text{New } u$$ will have in general a definite value. For $$r = 0,$$ the value of $$\bar u'$$ is evidently $$u.$$ We have, therefore,

98. If $$\text{Pot } \omega$$ has in general a definite value, Hence, by No. 71,  That is, If we set  we have where $$\omega_{1}$$ and $$\omega_{2}$$ are such functions of position that $$\nabla. \omega_{1} = 0,$$ and $$\nabla \times \omega_{2} = 0.$$ This is expressed by saying that $$\omega_{1}$$ is solenoidal, and $$\omega_{2}$$ irrotational. $$\text{Pot } \omega_{1}$$ and $$\text{Pot } \omega_{2},$$ like $$\text{Pot } \omega,$$ will have in general definite values.

It is worth while to notice that there is only one way in which a vector function of position in space having a definite potential can be thus divided into solenoidal and irrotational parts having definite potentials. For if $$\omega_{1} + \epsilon, \, \omega_{2} - \epsilon$$ are two other such parts, Moreover, $$\text{Pot } \epsilon$$ has in general a definite value, and therefore Rh