Page:The principle of relativity (1920).djvu/136

 the expression within the bracket may be written as

= ω_{h}[sum][part]νω_{k}/[part]x_{k} + ν[sum]ω_{k}[part]ω_{h}/[part]x_{k}.

The first sum vanishes in consequence of the continuity equation (b). The second may be written as

([part]ω_{h}/[part]x_{1})(dx_{1}/dτ) + ([part]ω_{h}/[part]x_{2})(dx_{2}/dτ) + ([part]ω_{h}/[part]x_{3})(dx_{3}/dτ) + ([part]ω_{h}/[part]x_{4})(dx_{4}/dτ)

= dω_{h}/dτ = (d/dτ)(dx_{h}/dτ)

whereby (d/dτ) is meant the differential quotient in the direction of the space-time line at any position. For the differential quotient (12), we obtain the final expression

(14) [integral][integral][integral][integral] ν(([part]ω_{1}/[part]τ)ξ_{1} + ([part]ω_{2}/[part]τ)ξ_{2} + ([part]ω_{3}/[part]τ)ξ_{3} + ([part]ω_{4}/[part]τ)ξ_{4})

dx dy dz dt.

For a virtual displacement in the sichel we have postulated the condition that the points supposed to be substantial shall advance normally to the curves giving their actual motion, which is λ = 0; this condition denotes that theξ_{h} is to satisfy the condition

w_{1}ξ_{1} + w_{2}ξ_{2} + w_{3}ξ_{3} + w_{4}ξ_{4} = 0.?] (15)

Let us now turn our attention to the Maxwellian tensions in the electrodynamics of stationary bodies, and let us consider the results in § 12 and 13; then we find that Hamilton's Principle can be reconciled to the relativity postulate for continuously extended elastic media.