Now, whilst a is less than CD, there will be no intersection, and hence no solution at all; and when a = CD, there will be one solution, the triangle being right angled. When a is less (in Fig. 1) or greater (in Fig. 2) than either CA or CA' (but greater than CD), there will be two intersections b and b' (Prop. x., Cor. 4), and hence two triangles ABC and AB'C, which have all the data, and therefore two solutions. When a is equal to, or greater than the less of the two CA, CA', but less than the other, there will be only one solution as ABC, or ABC. When a is greater than both CA and CA', the triangle that will be formed will not have the given angle A, but its supplement in the one case; and though formed with A in the other case, one of its sides will be greater than a semicircle, contrary to Prop. v. Whilst, therefore, the side a lies between b and Ē in value, there will be only one solution : but in all other cases, either two or none. Cor. The side b and its opposite angle B, in the case of the solution being unique, are of the same affection ; as is evident from Prop. X., Cor. 2. PROPOSITION XII. Two angles of a triangle and the side opposite to one of them being given, it is required to investigate the number of solutions of which a triangle constructed with these data admits. Let B be the given angle to which the given side BC is adjacent, and A the other given angle: it is required to find when two solutions, and when only one, will be compatible with the data. [In Fig. 1, B is acute, and in Fig. 2, B is obtuse.] Draw CR perpendicular to the side AB, and CQ perpendicular to CR: produce BC, BA, to meet again in B', make Qd = QB', and draw the great circle Cd. Then, since CQ, RQ are perpendicular to CR, Q is the pole of CR, and QR is a quadrant; wherefore (Prop. x.) CDR = CB'R = CBR. There will hence for any value of the angle A between CdR and CQR be two solutions, as a'BC and aBC; but for any value which shall give to A a position between d and B in Fig. 1, and between d and B’ in Fig. 2, there is only one solution as ABČ. Now the limit to the value of A on the right being Cdr, when A B, and the limit on the left being CBR' when A = t - B: and the progress from the one to the other being continuous, the condition of the solution being single, is, that A lies between B and T - B. When it is either less in Fig. 1, or greater in Fig. 2, than CQR, there is no solution ; and when between the limits B and Q, there will be two solutions. Whilst, therefore, the value of the angle A is between B and 7 B, there will be but one solution : but in all other cases, either two or none. Cor. The side b and its opposite angle B, in case of one solution, will be of the same affection, as is evident from Prop. x., Cor. 4. D H A PROPOSITION XIII. Spherical lunes and spherical wedges (Def. 10) are to one another as the spherical angles formed at the surface of the sphere by the enclosing arcs of great circles. Let the planes meeting in the diameter AA' of the sphere cut the surface in ACA', ABA', ADA'. Then the lune contained by ACA', ABA', will be to that contained by ABA', ADA', as the angle CAB to the angle BAD. For, let the plane CBD pass through the centre perpendicular to the diameter AA', cutting the planes which form the lunes in C, B, D. Take in this the arcs CE, EF, FG, etc., each equal to BC, and the arcs DH, HK, etc., each equal to BD; and through the axis AA', and the several points E, F, G, etc., and H, K, etc., draw planes cutting the surface of the sphere in the great circles AEA', AFA', AGA', etc., AHA', AKA', etc. Then it may be proved by super-position that the lunes BACA', CAEA', EAFA', FAGA', etc., are all equal; and likewise that BADA', DAHA', HAKA', etc., are all equal. Moreover, that the angles CAB, CAE, EAF, FAG, etc., which are equal to COB, COE, EOF, FOG, etc., and which are subtended by the equal arcs BC, CE, EF, FG, etc., are all equal; as likewise the angles BAD, DAH, HAK, etc., are all equal. Then reasoning precisely as in Pls, and Sols., Chap. II., Prop. x., the asserted conclusion follows at once. In the same manner it may be proved that the wedges are proportional to the angles at A. Cor. 1. The area of any lune is to the surface of the sphere as A to 27, A being the angle of the lune. Cor. 2. Also, since the area of the spherical surface is represented by 4 rów, we have the expression of the area of the lune to A, A A lune A 4 p2o 27 2 p?o. Of particular cases, notice should be taken of A T and A in which cases we have, severally, lune 90o = pew, and lune 180° 2 pw. It will be advantageous to write the mark ) for the word lune; in which case, the expressions above will be briefly expressed as follows: Ao 900: A B PROPOSITION XIV. The area of a spherical triangle is to the surface of the hemisphere as the excess of its three angles above two right angles is to four right angles. Let ABC be the spherical triangle, and complete the great circles which constitute its containing sides. Then the triangle A'B'C' is equal in all respects to ABC (Prop. ix.), and the hemisphere is equal to the four triangles A'BC, AB'C, A'B'C, and ABC or A'B'C'. But we have A A'BC = A) - A ABC A ABC. 2 A ABC = A) + B+C) - T. С , π) 2 w ; and hence, substituting and dividing by 2, we get A+B+C- 2 ση; which is identical in signification with the verbal enunciation of the theorem. COR. 1. Since the only variable quantity in the expression for the area of the spherical triangle is the spherical excess, E = A +B+C - To, the excess is a correct representation of the area in terms of the unit Cor. 2. A polygon of n sides being divisible into n - 2 triangles, , and the area of each triangle being expressible by means of the spherical excess, we shall have, if signify the sum of all the angles of the polygon, E, + E, +. En- . 2 w go? 2 o pl. 2 π Cor. 3. If on the surface of the sphere there be described P, polygons, one of which has n, sides, and so on to that which contains n, sides, (any two or more of these number of sides, however, may be equal); and if E1, E2, be the sums of the angles of the several polygons, P1, P2, Then 2,- (ni 2). TE 2, – (n, 2) π P 2 or, etc. 2 π P,: 2 w po?, 27 And by addition we have P. + P, + ...P, 20 27 Cor. 4. If the polygons P., P., ... Pp, together entirely and exactly cover the sphere, we shall have Σ + Σ, + En – (n, + n2 + ... N, – 2p) + 4 o polo 2 ore. 27 EXERCISES ON THE CONE, SPHERE, AND CYLINDER. 1. Equal circular sections are equally distant from the centre of the sphere; and those which are equally distant are equal. 2. The greatest section of the sphere is made by a plane through the centre; and that section which is nearer to the centre is greater than the more remote. And conversely, 3. If two spheres touch each other, the line joining their centres will pass through the point of contact. 4. Two spheres cannot touch one another in more than one point. 5. Through any point on the surface of a cone one tangent plane, and only one, can be drawn: and through any point witliout the surface, two. 6. If a sphere and right cone touch in two points, they touch in an entire circle. 7. Through any point on the surface of a right cone, innumerable tangent spheres may be described : but only one of them can touch it in more points than one, and this touches in a circle whose centre is in the axis of the cone. 8. If a plane cut a right cone and be not parallel to a tangent plane, then two spheres can be found (one on each side of the plane) which shall touch the plane in points and the cone in circles. 9. If two unequal spheres lie each wholly without the other, two right cones can be described which shall be tangential to the spheres; if the spheres cut each other, only one such cone can be described; and if one sphere lie wholly within the other, no such cone can exist, 10. Planes may be described to touch three spheres, cach of which lies wholly without the other two :-how many? Also, investigate the circumstances of any of the spheres cutting or lying within another. 11. Four unequal spheres lie each wholly without the others, and common cones are described to envelope each two: the vertices of these cones are situated in threes upon straight lines, and in sixes in four planes. CHAPTER IX. THE CONIC SECTIONS. The first volume of this course contains a short discussion on the conic sections by coordinates. So far as the investigation of properties is concerned, the coordinate method is by far the most simple and direct. The elements of the subject are then limited to the equations of the curves, and the equations of their tangents, normals, and diameters. In establishing that method, however, a geometrical property of the conic sections is assumed without proof. It is proposed, therefore, in this chapter, to show how those curves are formed on the surface of a cone, and thence to deduce by geometry some of their most important properties. This course will then contain the elements of the conic sections, both by geometry and analysis. DEFINITIONS. 1. A conic section is the figure formed by the intersection of a plane with the surface of a right cone. The cone is supposed to admit of indefinite extension, both below the circular base and beyond the vertex, so that the complete conical system is composed of two opposite cones, or two opposite sheets of the same cone. 2. A plane drawn through the vertex of the cone parallel to the cutting or sectional plane is called the directing plane. 3. The section receives different names, according to the position of the directing plane. When this plane touches the cone along one of its edges, the section made by the cutting plane is called the parabola. When the directing plane cuts both sheets of the cone, the figure is called the hyperbola. In this case there are two parts of the curve, which are often called opposite hyperbolas. In the investigation of properties, however, those parts are treated only as two branches of the same curve. When the directing plane cuts one sheet only of the complete cone, the figure is called the ellipse.f 4. The plane which passes through the axis of the cone, and is perpendicular to the sectional plane, is called the transverse plane. 5. The intersection of the transverse and sectional planes is called the transverse axis of the conic section. When this live meets the section in two points (as it evidently does in the ellipse and hyperbola), the portion of it intercepted between these points is called the transverse diameter; and when the transverse axis meets the section in one point * The sections made by a plane with the oblique cone, the ellipsoid, and some other surfaces, are also conic sections ; but it will be sufficient for the objects of this course to consider only the sections made with the right cone. † As varieties of these curves, the circle is a case of the ellipse, and is formed when the cutting plane is parallei to the base of the cone (Pls. 1. 24). The point is also another case when the cutting and directing planes coincide; also when the directing plane of the hyperbolic section coincides with the sectional plane, the section becomes two straight lines. |