Page:The principle of relativity (1920).djvu/134

 the value λ; these magnitudes are then functions of (x, y, z, t, λ). Let us now conceive of an infinitely thin space-time filament at the point (x y z t) with the normal section of contents dJ_{n} and if dJ_{n} + δdJ_{n} be the contents of the normal section at the corresponding position of the varied filament, then according to the principle of conservation of mass—(ν + dν being the rest-mass-density at the varied position),

(8) (ν + δν) (dJ_{n} + δdJ_{n}) = νdJ_{n} = dm.

In consequence of this condition, the integral (7) taken over the whole range of the sichel, varies on account of the displacement as a definite function N + δN of λ, and we may call this function N + δN as the mass action of the virtual displacement.

If we now introduce the method of writing with indices, we shall have

(9) d(x_{h} + δx_{h}) = dx_{h} + [sum]k [part]δx_{h}/[part]x_{k} + [part]δx_{h}/[part]λ dλ

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

Now on the basis of the remarks already made, it is clear that the value of N + δN, when the value of the parameter is λ, will be:—

(10) N + δN = [integral][integral][integral][integral] ((νd(τ + δτ))/dτ)dx dy dz dt,

the integration extending over the whole sichel d(τ + δτ) where d(τ + δτ) denotes the magnitude, which is deduced from

[ sqrt](-(dx_{1} + dδx_{1})^2 - (dx_{2} + dδx_{2})^2 - (dx_{3} + dδx_{3})^2 - (dx_{4} + dδx_{4})^2)

by means of (9) and

dx_{1} = ω_{1} dτ, dx_{2} = ω_{2} dτ, dx_{3} = ω_{3} dτ, dx_{4} = ω_{4} dτ, dλ = 0