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a constant, are proportional to (df), (df1⁄2), (dfa);

and

those of a curve of the second system, when έ = a constant, are

proportional to (d), (da), (d): hence it follows that if u is

dn dn

the angle at which two curves of the systems intersect at (§,n),

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the last equivalent of being expressed in the notation of determinants.

Let 0 and ' be the angles between ds and the lines at (, n) of the first and second systems respectively; then

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Also to determine the direction-cosines of the normal to the surface at (,n) we have

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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,

= 0; (103)

(df) (df) + (df) (df) + (da) (d) =

in which case,

dn

dn

ds2 Edg2+G dn2.

(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 έ+d§, 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,

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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.

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whence, squaring and adding, and substituting from (30), we

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As this formula is general, it includes all the preceding; and the following are examples of its application.

f

Ex. 1. Let x = &, y = n, z = f (x, y) = ƒ (§, n).

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Ex. 2. The equations to the skew helicoid are x = έ cos n, y = έsin ŋ, z = kaŋ.

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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 7 are a and 0, and 2 and 0 respectively so that if a = the corresponding area,

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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 έ and ʼn by the equations x =ƒ1 (§, n), y = ƒ2 (§, n); then we have, as in (34) Art. 213,

dx dy = |(df), (df)|dŋ dɛ,

=

dn

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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 έ and 7, and P, to be η, P3 the point of intersection of the lines corresponding to §+d§, n+dn, so that PP2, P1 P3 are two consecutive lines of the first system, and PP1, P2 P3 are two consecutive lines of the second system: then, in terms of curvilinear coordinates, p is (§, n), P1 is (§+d§, n), P2 is (§, n+dn), P3 is (§+d§, n+dn); and in terms of rectangular coordinates P is (x,y), P, is ( x +

(dc) dɛ,y+ dɛ

(dy) aɛ), 2, is (x + (de) de + (dz) dn, y + (d) dε + (d) dn), and F, is

dx

(x+ (dr) dn, y+ (d) dn). Hence as the area of a quadrilateral

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.~, y,, if da is the area of the surface-element whose angu

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If the area-element, delineated in the figure, is on a curved surface, it is evident that PP, and PP2 are the lines denoted by do and do' in Art. 247, and by a similar process we have

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η,

250.] The equations, which connect x, y, z with έ and ŋ, have been taken in (77) to be explicit. Suppose them however to be implicit, and of the forms

F(x, y, z, §, n) = 0, F2(x, y, z, §, n) = 0, F ̧ (x, y, z, §, n) = 0; (117) then by differentiation we have, making obvious substitutions, à1dx+b1dy + c1dz + a1 d§ + ß1 dŋ = 0,

1

a dx + b2 dy + c2 dz + aq d§ + ß2 dŋ = 0,

dn

az dx + bzdy + cz dz + az d§ + ẞ3 dŋ = 0;

C3

(118)

But if r (x, y, z) = 0 is the equation to the surface, it results by the elimination of έ and ŋ from (117); and as hereby x, y, z are connected by a relation, we have also

η

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This relation between dx dy and dz will also arise by the elimination of de and dy from (118). Now from (118) we have

|a, b2, cgdx = {a1 | b2, C3 | + a2 | b3, C1 | + a3 | b1, C2| } d§ − |

=

— {ß1| b2, C3|+B2| b3, C1 | + B3| b1, C2|} dn; (120) -|α, b2, c3 dε-|ẞ1, b2, C3 | dn;

(121)

and also similar values for dy and dz. These equations are of the forms (86); and consequently all the theorems which have been deduced from (86) may, mutatis mutandis, be deduced from these equations.

Y y

PRICE, VOL. II.

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