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n = a constant, are proportional to (oli), , ( ); and those of a curve of the second system, when & = a constant, are proportional to Cent), Cant), ): hence it follows that if u is
"on" on the angle at which two curves of the systems intersect at ($,n), Cosw = =
the last equivalent of v being expressed in the notation of determinants.
Let 0 and 6' be the angles between ds and the lines at (6, n) of the first and second systems respectively; then
Ed&+Fdn cos 0 =
(98) ds Et
Also to determine the direction-cosines of the normal to the surface at ($, n) we have
udx + v dy + w dz = 0;
which give the direction-cosines of the normal.
If every one of either of these systems of lines intersects at right angles every one of the other system so that the two systems are orthogonal, then the surface becomes divided into a series of infinitesimal rectangles, and w = 90°; and consequently from (87), F = 0; that is,
in which case, ds2 = Edg? +G dna.
(104) Also hereafter we shall find that this expression admits of further simplification, by taking such a system of lines, that either E or G is equal to unity.
248.] Although the substance of the preceding article has reference chiefly to length-elements of curves on surfaces, and thus to the rectification of curves of double curvature, yet it arises out of the necessary explanation of the system of coordinates; and it will be moreover of considerable use in the sequel. Now we return to the proper subject of the present chapter, and shall investigate the most general analytical value of the surfaceelement in terms of curvilinear coordinates. Imagine two systems of curves, as explained above, to be described on the surface; the surface will thereby be divided into elements, an element being contained between two curves of the first system, viz. $, and $ + dę, and between two curves of the second system, viz. n, and n+dn. This element will be approximately a parallelogram, and the sides of it, which meet at the point ($, n), will be do and do', the length of which have been determined in (90) and (91); and w, as determined in (93) or (94), will be the angle between them; so that if da is the surface-element,
da = do doʻsin w
employing the value of v as given by the determinant-notation in (97).
This value of the surface-element may also be found by analytical transformation according to the principles of chap. VIII.
whence, squaring and adding, and substituting from (30), we have
which is the same result as (106).
As this formula is general, it includes all the preceding; and the following are examples of its application.
Ex. 1. Let x = 5, y = 9, z = f (x, y) = f(€, n). : ods) = 1, carte ) = 0, Galia) = (45) = (.5),
which is the expression already determined.
Ex. 2. The equations to the skew helicoid are x = $ cos n, y= & sin n, z = kan.
Now this surface is that of the under-side of spiral staircases ; thus if the radius of the cylinder which incloses the staircase is a, and the surface of one turn is required, the limits of & and n are a and 0, and 27 and 0 respectively : so that if a = the corresponding area,
249.] The preceding theory of curvilinear coordinates gives a geometrical explanation of the general transformation of a double integral. For if the double integral involves two variables x and y; and if these are connected with new variables & and n by the equations x = fi($, n), y=fz($, n); then we have, as in (34) Art. 213, då dy = \(La), Cen) don dę,
Now the left-hand member of (111) represents the infinitesimal area-element referred to rectangular coordinates, and therefore the right-hand member expresses the analogous element in terms of curvilinear coordinates. Suppose then in fig. 43 P to be the point of intersection of the lines corresponding to & and n, and Pg to be the point of intersection of the lines corresponding to $+de, n+dn, so that PP2, P, Pg are two consecutive lines of the first system, and PP1, P2P, are two consecutive lines of the second system : then, in terms of curvilinear coordinates, p is (, n), P, is (5+dę, n), P, is (, n+dn), P, is ($+de, n+dn); and in terms of rectangular coordinates P is (2,9), P, is (x+ (*) d£,y+ () ),
, ysl, if da is the area of the surface-element whose angu. lar points are P, P1, P3, P2,
If the area-element, delineated in the figure, is on a curved surface, it is evident that PP, and PP, are the lines denoted by do and do' in Art. 247, and by a similar process we have