IN the third edition I have made some additions which I hope will be found valuable. I have considerably enlarged the discussion on the connexion of Formulæ in Plane and Spherical Trigonometry; so as to include an account of the properties in Spherical Trigonometry which are analogous to those of the Nine Points Circle in Plane Geometry. The mode of investigation is more elementary than those hitherto employed; and perhaps some of the results are new. The fourteenth Chapter is almost entirely original, and may deserve attention from the nature of the propositions themselves and of the demonstrations which are given. CAMBRIDGE, July, 1871. IV. Relations between the Trigonometrical Functions of the Sides SPHERICAL TRIGONOMETRY. I. GREAT AND SMALL CIRCLES. 1. A SPHERE is a solid bounded by a surface every point of which is equally distant from a fixed point which is called the centre of the sphere. The straight line which joins any point of the surface with the centre is called a radius. A straight line drawn through the centre and terminated both ways by the surface is called a diameter. 2. The section of the surface of a sphere made by any plane is a circle. Let AB be the section of the surface of a sphere made by any plane, O the centre of the sphere. Draw OC perpendicular to the plane; take any point D in the section and join OD, CD. Since OC is perpendicular to the plane, the angle OCD is a right angle; therefore CD=√(OD- OC). Now O and C are fixed points, so that OC is constant; and OD is constant, being the radius of the T. S. T. B sphere; hence CD is constant. Thus all points in the plane section are equally distant from the fixed point C; therefore the section is a circle of which C is the centre. 3. The section of the surface of a sphere by a plane is called a great circle if the plane passes through the centre of the sphere, and a small circle if the plane does not pass through the centre of the sphere. Thus the radius of a great circle is equal to the radius of the sphere. 4. Through the centre of a sphere and any two points on the surface a plane can be drawn; and only one plane can be drawn, except when the two points are the extremities of a diameter of the sphere, and then an infinite number of such planes can be drawn. Hence only one great circle can be drawn through two given points on the surface of a sphere, except when the points are the extremities of a diameter of the sphere. When only one great circle can be drawn through two given points, the great circle is unequally divided at the two points; we shall for brevity speak of the shorter of the two arcs as the arc of a great circle joining the two points. 5. The axis of any circle of a sphere is that diameter of the sphere which is perpendicular to the plane of the circle; the extremities of the axis are called the poles of the circle. The poles of a great circle are equally distant from the plane of the circle. The poles of a small circle are not equally distant from the plane of the circle; they may be called respectively the nearer and further pole; sometimes the nearer pole is for brevity called the pole. 6. A pole of a circle is equally distant from every point of the circumference of the circle. Let be the centre of the sphere, AB any circle of the sphere, C the centre of the circle, P and P' the poles of the circle. Take any point D in the circumference of the circle; join CD, OD, PD. Then PD = √(PC2+CD3); and PC and CD are constant, therefore PD is constant. Suppose a great circle to pass through the points P and D; then the chord PD is constant, and therefore the arc of |