I. Great Circle.-The boundary of every section of a sphere made by a plane passing through the centre.

II. Small Circle. The boundary of every section of a sphere made by a plane not passing through the centre.

III. Poles of a Circle.-The extremities of that diameter of the sphere which is perpendicular to the plane of the circle.

IV. Spherical Triangle. The portion of the surface of a sphere contained by three arcs of great circles which cut one another two and two.

V. Lune.-The spherical surface included between two semicircles containing an angle 6.

FUNDAMENTAL FORMULE.

When, A, B, C denote the angles of a spherical triangle ▲ a, b, c denote the sides respectively opposite.

1.

cos a = cos b cos c + sin b sin c cos A.

Solution of Right-Angled Triangles.
C = 90°, c = hypothenuse.

Circular parts, a, b, 90 - c, 90 — A, 90 — B.

18. Legendre's Theorem. If A'B'C' are angles of a plane triangle whose sides are of the same length as those of a

corresponding spherical triangle, and small compared with the radius of the sphere; then—

I. Co-ordinate axes.-Two straight lines intersecting each other in a point called the origin, to which the position of a point in the same plane with them is referred. The axes are rectangular or oblique, according as they intersect each other at right angles or obliquely.

II. Ordinate of a point.-The distance of the point from one of the axes, measured parallel to the other axis.

III. Abscissa of a point.-The part of the axis cut off by the ordinate.

IV. Co-ordinates.-The abscissa and ordinate.

V. Radius vector of a point.-Its distance from a fixed point called the pole.

VI. Vectorial angle or angle of revolution.-The angle which the radius vector makes with a fixed line called the initial line.

VII. Polar co-ordinates of a point.-The radius vector and vectorial angle.

VIII. Equation to a line. The equation which expresses generally the relation between the co-ordinates of any point whatever in the line.

IX. Locus of an equation.-The line which contains all the points determined by the proposed equation.

radius vector
vectorial angle,,

(xy) denotes the point whose co-ordinates are x and y.

FORMULE FOR TRANSFORMATION OF

CO-ORDINATES.

When h denotes the abscissa of a new origin.

ordinate of a new origin.

angle between two oblique axes.

the axes of x.

the axes of y.

2. For change of direction of axes, both systems rectangular,

a becomes a cos a y sin a,

y

becomes a sin a + y cos a.

3. For change of rectangular to oblique co-ordinates,

a becomes a cos a + y sin ß,

y

becomes a sin ay cos B.

4. For change of oblique to rectangular co-ordinates,

5. For change of rectangular to polar co-ordinates,

a becomes r cos 0,

y becomes r sin 0.