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If A = B = 5, the triangle becomes half of a lune; and consequently the area of the lune = 2 aể c.

If c = 27, the area of the whole sphere = 47a2.

Ex. 2. To find the definite integral expressing the surface of the ellipsoid in terms of polar coordinates.

Let the equation to the ellipsoid be given by the system (38); then

x = a sin a cos B = r sin 0 cos ,
y = b sin a sin ß = r sin 6 sin , }

(72) z = c cos a = r cos 0; so that eliminating r,

a tan a cos ß = c tan 6 cos , ?

6 tan a sin ß = c tan 6 sind ;)
i ab (sec a)? tan a da = c2 (sec 6)2 tan 0 do do.
.: abc sin a da = 73 sin 0 do do.

(73) and substituting in (41),

pipgo3 sin 0 do do. A=f4***

(74) J- Jop which is the required expression. Also for the whole ellipsoid

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= 47 abc.

(75) Ex. 3. The lemniscata, pe=asin 20, is described in the plane of (x,y); and in planes perpendicular to that of (x,y) circles are described on the radii vectores as diameters; find the area of the surface.

The equation to the surface is (m2 + y2 + z2)2 = 2 a Xy; and in polar coordinates po2 = a(sin 0)2 sin 20; whence, by (64),

da = a(sin 6)2 do do ; and taking the surface described on one loop,

A = 2 /"S*a* (sin 6)? do do

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Ex. 4. If p is the perpendicular from the origin on the tangent plane at the point of the surface-element, and if the surface is closed, and has no singular points,

po * po3 sin 0 do do

SECTION 5.-Gauss' System of Curvilinear Coordinates *.

245.] In Art. 193, Vol. I, it is remarked that the equation to a plane curve may be expressed in terms of a subsidiary angle, or indeed in terms of any subsidiary quantity, by means of two equations; so that these two equations taken simultaneously express the curve; and the original equation to the curve is found by the elimination of this subsidiary quantity; and this system of reference has been applied in this and the preceding chapters to the rectification of curves and the quadrature of plane surfaces. Now the same principle is also applicable to curved surfaces; for as the equation to a surface involves three variables, viz., X, Y, Z, so in the general case each of these coordinates must be a function of two subsidiary and independent quantities; and the equation to the surface will arise from the elimination of these quantities from the given equations. Let & and n be these subsidiary quantities; and let the equation to the surface in terms of x,y,z be F(x, y, z) = 0 :

(76) and let x, y, z be connected with g, n by the equations,

x = fi($, n), y = fr($, n), z = f ($, n); (77) where fi, fa, fg are functions such that f (x,y,z)=0 arises by the elimination of & and n from them.

The following are particular cases of these equations.

Ex. 1. The ellipsoid may be expressed by the following equations ; viz.

X = a sin n cos , y = b sin n sin ļ, 2 = c cos n; (78) because if we eliminate & and n, we have 1. x2 2 22


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which is the ordinary equation to the ellipsoid.

* The matter of the following section is in a great measure taken from Gauss' celebrated Memoir entitled “ Disquisitiones generales circa superficies curvas,” which is contained in Vol. VI of the Memoirs of the Royal Society of Sciences of Gottingen, 1828. It has been reprinted as an Appendix to Monge's Application d’Analyse &c., edited by M. Liouville, Bachelier, Paris, 1850; and Liouville has added some notes elucidating various parts of the system in its application to curvature and to geodesic lines. Also the student desirous of further information may consult a profound paper of M. Ossian Bonnet on the General Theory of Surfaces, in the Journal de l'Ecole Polytechnique, Cahier XXXII, Paris, 1848.


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If a=b=c, the ellipsoid becomes a sphere: in which case (78) become

x = a sin n cos s, y = a sin n sin ę, z= a cos n; (80) and g, n are the ordinary polar coordinates.

Ex. 2. The hyperboloid of one sheet may be expressed by the following equations ; viz.

x = a sec n cos &, y = b sec n sin , z = c tan n; (81) because if we eliminate & and n, we have

X2 22 22
2 + 12 - = 1,

(82) which is the ordinary equation to the surface.

Ex. 3. The skew helicoid, (88), Art. 368, Vol. I, may be expressed by the equations

x = & cos n y = $ sinn, z = kan. (83) 246.] These examples are sufficient to illustrate the method; and we return to the general equations (76) and (77). As & and n are independent variables, we may consider them separately. Firstly let n have a determinate value, say n = c; and let & vary; and let & be eliminated from the equations (77); then two equations result in terms of x,y,z and n, each of which represents a surface; and as they are simultaneous, taken together they represent a line, which is generally a curve of double curvature; and which because of the simultaneity of all the equations, evidently lies on the surface (76); this line is called n= cz; and by giving different values to cz, we obtain a series of such lines, all lying on the surface (76). Secondly let & have a determinate value, say $ = c; and let n vary; then, as before, it is evident that the equations formed by the elimination of n will represent a line lying on the surface (76); and if different values are given to cy, we shall have a series of lines lying on the surface. Thus the two equations § = ly, n= C2,

(84) taken separately, represent two different systems of curves drawn upon the surface (76). Now, as the systems are continuous, every point on the surface will be at the intersection of two curves, one of which is a member of the first, and the other is a member of the second system : and the point is determined whenever these curves are given. Suppose (xo, Yo, z.) to be a point on the surface, corresponding to which the values of & and n are &o and no, these latter quantities admitting of the value zero; then the intersection

of so and n, may conveniently be taken to be the origin ; and any other point to be at the intersection of two lines corresponding to given values of f and n. And & and n may fitly be called the curvilinear coordinates to that point.

Thus take the ellipsoid, as given by the equations (78). Let be eliminated : then we have

22 22
+ 12 = (sin n)?, 2 = c cos n;

(85) now for a given value of n the former of these equations represents an ellipse in the plane of (x, y), and generally an elliptical cylinder whose axis is the z-axis; and as the latter represents a plane parallel to the plane of (x, y), it appears that these equations represent an ellipse drawn on the surface of the ellipsoid, and lying in a plane parallel to that of (x, y). Thus the second system of lines as given in (84) is a series of ellipses described on the surface of the ellipsoid in planes perpendicular to the z-axis. Again, if we eliminate n from (78), we have

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which represents a series of planes passing through the z-axis; and as these intersect the ellipsoid in ellipses, the first system of lines as given in (84) is a series of ellipses described on the surface by planes passing through the z-axis. Thus a given point on the surface of the ellipsoid will be determined by means of the two ellipses, one of each system, which intersect at that point. The particular values of E and n which determine these ellipses are the curvilinear coordinates to the point.

247.] Let there be two points (x, y, z) (x+dx, y+dy, z+dz) on the surface F(x, y, z) = 0 infinitesimally near to each other; and let us suppose the first to be at the intersection of & and n, and the second at the intersection of $+d and n+dn; and let ds be the distance between them; let us investigate the relations between these points in terms of g, n and their differentials. Differentiating (77) we have

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dgå = dx2 +dya +dza
= Edf? +2 rde dn+adn;

(88) and if s is the length of a curve described on the given surface between the points (&1, N) and (fo, no),

8 = [ {Edf? +2f dn+G dn2}t; (89) 1 and 0 being the subscript letters of, and thus conveniently used as abbreviated expressions for, the limits of the integral. I may observe that this is a notation which will be frequently employed hereafter.

Equation (88) enables us to give a simple geometrical interpretation to E, F and g; let dn= 0, that is, let us suppose , to be constant, and let us pass from ($, n) to ($+, n). Then if do is the distance between these points, it is plain from (88) that, do = Et df;

(90) hence et de is an infinitesimal arc-element of the curve n = Cz, intercepted between two consecutive curves of the other system. Similarly, if do' is the value of ds, when & = C, do' = głdn;

(91) and consequently Gł dn is an infinitesimal arc-element of the curve $ = C1, intercepted between two consecutive curves of the other system.

Let w = the angle at which do and do are inclined to each other; let ds be the distance between ($, n) and (f+df, n+dn); then

ds? = do + 2 do do' cos w+ do'

= E ? +2 et al dn cos w + Q dna ; (92) and equating this to the value given in (88), we have

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Or thus; the direction-cosines of a curve of the first system, when

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