and by adding these equations together after having developed (u — v)”, and (v-u)", we have

and making the same substitutions as before in virtue of (3), and recollecting that, because n is even, (✔—1)" = 1, the upper sign having place when n is either of the numbers 2, 6, 10, &c. and the lower sign when n is either of the numbers 4, 8, 12, &c. we have for the development of sin."r

and developing (u — v)n, (v — u)" as before, and taking the sum of these equations, we have

But from the equations (2),

un —v2 = 2 sin. nx √ = 1, u2 v2 = 1;

n- =

consequently, since (1) 1, the foregoing development becomes

the upper sign having place when n— 1 is either of the numbers 2, 6, 10, &c. and the lower sign having place when n- 1 is either of the numbers 4, 8, 12, &c. The general term of the first series of numbers is 4m+2, that of the second series 4m.

END OF PART I.

་་

PART II.

ELEMENTS OF SPHERICAL TRIGONOMETRY.

СНАРТER I.

ON THE SPHERE.

(34.) A SPHERE is a solid whose surface is every where equally distant from a certain point within it, called the centre. It may be generated by the revolution of a semicircle about the diameter.

Any line drawn from the centre to the surface of the sphere is called the radius; and the line through the centre having both its extremities in the surface is the diameter.

A plane surface, or simply a plane, is that in which if any two points whatever be taken, the straight line which joins them shall lie wholly in that surface.

A plane may be drawn through any three points, taken at random in space, but not through more than three; for having joined two of the proposed points by a straight line we may pass a plane through this line in any direction, and we may turn it round upon this line till it arrives at the other point. Three points, therefore, not in the same straight line, fix the position of a plane.

It follows from this that the common intersection of two planes must be a straight line; for, if among the points in the intersection there be three which are not in the same straight line, the two planes passing through them must coincide and form but one.

A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line in that plane, drawn through its foot,

or the point where the perpendicular meets the plane. These definitions will suffice for the purpose of establishing the necessary preliminary theorems of spherical Geometry.

(35.) If a sphere be any how cut by a plane, the section must be a circle.

Let C be the centre of the sphere, and ADB the plane section; draw Cc perpendicular to this plane, and from c draw any line cD in the section and terminating at the surface; then the angle CCD must be a right angle. Join H CD, then wherever the point D may be, CD will always be of the same constant length, being the radius of the sphere; and in conse

B

quence of the right angle c, cD = CD2-Cc2; hence cD must have the same constant length in whatever direction it be drawn, that is, the bounding line ADB is the circumference of a circle of which c is the centre.

The circle is, obviously, the larger, as it is nearer to the centre C of the sphere, or as its perpendicular distance Cc is less, because CD being constant cD increases as Ce diminishes, and becomes the greatest possible when Cc is 0, that is, when the section passes through the centre of the sphere; hence every circle whose plane passes through the centre of the sphere is called a great circle of the sphere, and every other a small circle.

It is obvious that the circumference of a great circle may be drawn through any two points on the surface of a sphere, because a plane may be drawn through these two points and through the centre also, but a great circle cannot be drawn through three points on the surface, taken at random, because then a plane might be drawn through four points taken at random; a circle of some kind, however, may always be drawn through three points on the surface of the sphere, since a plane may be drawn through them.

The line Cc from the centre of the sphere perpendicular to the plane of the circle passes, as we have seen, through its centre c; if this line be produced both ways to the surface of the sphere, the opposite points P, P', are called the poles of the circle. Thus every circle on the sphere has two poles diametrically opposite, the diameter which joins them