To find the simplest cases, I shall seek first an expression for manifoldnesses of n - 1 dimensions which are everywhere equidistant from the origin of the linear element; that is, I shall seek a continuous function of position whose values distinguish them from one another. In going outwards from the origin, this must either increase in all directions or decrease in all directions; I assume that it increases in all directions, and therefore has a minimum at that point. If, then, the first and second differential coefficients of this function are finite, its first differential must vanish, and the second differential cannot become negative; I assume that it is always positive. This differential expression, of the second order remains constant when ds remains constant, and increases in the duplicate ratio when the dx, and therefore also ds, increase in the same ratio; it must therefore be ds2 multiplied by a constant, and consequently ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx, in which the coefficients are continuous functions of the quantities x.
ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx
(1) “Let us suppose a variable piece of a manifoldness of one dimension” - I’m not sure what work the word ‘variable’ does here in ‘variable piece of a manifoldness’. Can one take an invariable piece of a manifold? And what would this distinction mean? — StreetlightX
(2) “Let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness.” - Here, I’m not sure what work ‘not constant’ is doing. Is it the variation of position on the manifoldness we are asked to think of is ‘not constant’? — StreetlightX
For this purpose let us imagine that from any given point the system of shortest limes going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin.
It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. Let us introduce now instead of the dx0 linear functions dx of them, such that the initial value of the square of the line-element shall equal the sum of the squares of these expressions, so that the independent varaibles are now the length s and the ratios of the quantities dx. Lastly, take instead of the dx quantities x1, x2, x3,..., xn proportional to them, but such that the sum of their squares = s2
When we introduce these quantities, the square of the line-element is \sum dx^2 for infinitesimal values of the x,
. Let us introduce now instead of the dx0 linear functions dx of them, such that the initial value of the square of the line-element shall equal the sum of the squares of these expressions, so that the independent varaibles are now the length s and the ratios of the quantities dx.
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