Page:The principle of relativity (1920).djvu/135

 therefore:—

(11) (d(τ + δτ))/dτ = [sqrt]( -[Sum](ω_{h} + [Sum]([part]δx_{h}/(part]x_{k})ω_{k})^2)

[ k = 1, 2, 3, 4. [ h = 1, 2, 3, 4.

We shall now subject the value of the differential quotient

(12) ((d(N + δN)/dλ)

(λ = 0)

to a transformation. Since each δx_{h} as a function of (x, y, z, t) vanishes for the zero-value of the paramater λ, so in general dδx_{k}/([part]x_{h} = 0, for λ = 0.

Let us now put ( [part]δx_{h}/[part]λ) = ξ_{h} (h = 1, 2, 3, 4) (13) λ = 0

then on the basis of (10) and (11), we have the expression (12):—

= -[Integral][Integral][Integral][Integral][Sum]ω_{h}(([part]ξ_{h}/[part]x_{1})ω_{1} + ([part]ξ_{h}/[part]x_{2})ω_{2} +([part]ξ_{h}/[part]x_{3})ω_{3} + ([part]ξ_{h}/[part]x_{4})ω_{4})

dx dy dz dt

for the system (x_{1} x_{2} x_{3} x_{4}) on the boundary of the sichel, (δx_{1} δx_{2} δx_{3} δ_{4}) shall vanish for every value of λ and therefore ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4} are nil. Then by partial integration, the integral is transformed into the form

[Integral][Integral][Integral][Integral][Sum]ξ_{h}([part]νω_{h}ω_{1}/[part]x_{1} + [part]νω_{h}ω_{2}/[part]x_{2} + [part]νω_{h}ω_{3}/[part]x_{3} + [part]νω_{h}ω_{4}/[part]x_{4}) dx dy dz dt