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The condition that p, a, ß may terminate in the same line is

Pp+qa+roß = 0,
subject to the identical relation

P+q + r = 0.

Similarly

pp+qa++8y = 0,

with

P+9+r+8 = 0,

is the condition that the extremities of four vectors lie in one plane. § 30.

Examples with solutions. § 31.

Differentiation of a vector, when given as a function of one number. SS 32–38.

If the equation of a curve be

p= $ (8)

where s is the length of the arc, dp is a vector tangent to the curve, and

its length is ds. SS 38, 39.

Examples with solutions. $$ 40-44.

EXAMPLES TO CHAPTER I.

22-24

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Here we begin to see what a quaternion is. When two vectors are parallel

their quotient is a number. SS 45, 46.

When they are perpendicular to one another, their quotient is a vector per-

pendicular to their plane. $ 47, 72.

When they are neither parallel nor perpendicular the quotient in general

involves four distinct numbers—and is thus a QUATERNION. § 47.
A quaternion regarded as the operator ich turns one vector into another.

It is thus decomposable into two factors, whose order is indifferent, the
stretching factor or TENSOR, and the turning factor or VERSOR. These

are denoted by Tq, and Uq. § 48.

The equation

B = qa

B

gives = 9, or Ba-1 =%, but not in general

a-?ß = 'q. 49.

q or Ba-l depends only on the relative lengths, and directions, of B and a.

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K(gr) = Kr. Kq. $ 55.

Proof of the Associative Law of Multiplication

p.qr = Pq.1. SS 57-60.

[Digression on Spherical Conics. § 59*.]

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Ka= -a,

that of perpendicular vectors, a vector; if quaternions are to deal with

space indifferently in all directions. 893.

EXAMPLES TO CHAPTER II.

46, 47

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CHAPTER VIII.-SURFACES OF THE SECOND ORDER .

135-151

EXAMPLES TO CHAPTER VIII.

151-154

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