three consecutive values: and two of the numerators are imaginary, unless a = b = c in which case both = 0.

65. If a, ß, y... be the n roots of the equation

...

66. If p and n be any integers and W1, W21 wn-1 be the nth roots of unity, excluding unity itself, then the remainder when p is divided by n is

67. By making x complex in the expansion of sin x into factors, show that

68. From the equation log sec (x + ) = log sec x + ô tan x +

69. Hence since log (2x cosec 2x) – log (x cosec x) = log secx,

CHAPTER XXII.

GEOMETRICAL INTERPRETATION OF IMAGINARIES.

VECTORS.

665. IN Chap. XII. the symbol (LM) was used to represent the line LM, regarded as drawn from L to M. It was called a directed length.

In that chapter, the operation of adding directed lengths in the same line was explained so as to lead to the formula

(LM) = (LT) + (TM)

for any points L, M, T in the same line.

A directed length could thus be represented by an algebraical number, affected by a plus or minus sign.

666. We now extend the use of directed lengths, so as to combine in operation directed lengths in different directions.

Here our symbols are bound down to a particular direction in space as well as to a particular length.

The symbols in Chap. XII. should rather have been called (merely) affected lengths than directed lengths: but Vector is the term used to express directed lengths in the extended sense to be now explained.

667. We may give the following definition :

DEF.

The position of any one point in space with respect to any other is called a Vector.

Now the position of any point B with respect to any point A is determined by (1) the length, (2) the direction of the line AB.

The vector involves, therefore, both length and direction.

The symbol (AB) may be extended to represent the vector AB.

668. If the line AB moves always parallel to itself, then its vector remains the same.

In other words, if (AB) and (CD) are equal, parallel, and in the same sense,

669.

Vector (AB) Vector (CD).

The extended use of directed lengths will of course involve the formulæ :

670. In Art. 340, it was shown that if A, B, C be any three points in space, the sum of the projections of (AB) and (BC) upon any line in the plane ABC is equal to the projection of (AC) upon that line.

For this reason it is convenient to define addition of vectors by the equation

(AB) + (BC) = (AC').

This of course includes the special case when A, B, C are in the same line. In other cases, the equation cannot be directly interpreted arithmetically, but only indirectly: i.e. by projecting upon any line in the plane ABC.

671. The above addition of vectors shows that the sum of two vectors drawn to and from a common point is the vector drawn from the initial point of the first to the final point of the second.

It follows at once that

The vector-sum of the sides of any closed figure, directed from point to point round the figure, is zero.

This is equivalent to Prop. II. Art. 341.

672. If two vectors are expressed by lines drawn from the same point, e.g. (AB), (AD); then the sum of (AB) and (AD) is found by drawing (BC) equal, parallel, and in the same sense as (AD), so that we have

The sum of two vectors, drawn from the same point, is a vector drawn from that point and represented by the diagonal of the parallelogram of which the two given vectors are adjacent sides.

673. If three non-coplanar axes be drawn from any point O, any vector (OP) can be resolved into the sum of three vectors in the direction of these axes.

For, let OX, OY be any two axes in the plane of the paper: and OZ any axis not in this plane. Then from any point P we may draw PN parallel to OZ, cutting the paper in N. And from N we may draw NM parallel to OY, cutting OX in M. Thus

(OP) = (OM) + (MN) + (NP),

where (OM) is along OX, (MN) parallel to OY, and (NP) parallel to OZ.

674. In the above result, the lengths of OM, MN, NP are entirely determined by the length and the direction of OP: and vice versa.

That is; The vector (OP) depends on three numbers.

Hence to equate one vector with another involves our equating each of three lengths with the corresponding parallel length.

675. If we are confined to a single plane, then any vector may be resolved into the sum of two vectors in the direction of any two axes drawn in the plane from a point.

Hence in this case to equate any one vector with another involves our equating each of two lengths with the corresponding parallel length.

Division of Vectors.

676. Since equal and parallel lines drawn in the same sense have equal vectors, we may represent any vector by a line drawn from an arbitrarily chosen point 0.

Again, since any two straight lines drawn from a point will lie on one plane, we may represent any two vectors by two lines (OA), (OB) drawn in the plane of the paper.

677. Draw (OA), (OB) in the plane of the paper to represent any two vectors.

Now a certain operation must be performed upon (04) in order to change it into (OB).

i.e.

This operation may be called the division of (OB) by (OA):

(OB)

(OA)

means the operation of changing (04) into (OB).

678. If (OA) and (OB) were in the same line and in the same sense the quotient of (OB) by (OA) would be the ratio of (OB) to (OA): i.e. the number by which the length of OA would have to be multiplied to give the length of OB.

In this particular case, then, the above definition of division would coincide with the ordinary definition.

679. But if (OA) and (OB) were in different lines, then we should have

(1) To increase or decrease the length of OA to the length of OB: and then

(2) To revolve the changed length through an angle AOB in the plane AOB.