Second proof of the associative law of multiplication, § 85.

Proof of the formulae

Saẞ=SBa,

Vaẞ= - VBa,
αβ = Κβα,

S.qrs=S.srq,

S.aBy=S.Bya=S.yaß=S.ayß= &c. §§ 86-89.

Proof of the formulae

V.a VBy=ySaẞ— BS'ya,

V.aBy=aSBY-BSya+ySaß,

Γ.αβγ = Γ.γβα,

d.S.aßy=aS.Byd + BS.yad + yS.aßd,

=3

VaßSyd+ VBySad + VyaSßd, §§ 90-92.

Hamilton's proof that the product of two parallel vectors must be a scalar,

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

with space indifferently in all directions, § 93.

EXAMPLES TO CHAPTER II. .

59, 60

CHAPTER III.-INTERPRETATIONS AND TRANSFORMATIONS OF

QUATERNION EXPRESSIONS.

If be the angle between two vectors, a and B, we have

61-85

is the volume of the parallelepiped three of whose conterminous edges

are a, 8, 7. Hence

Simple propositions in plane trigonometry, § 104.

Proof that aẞa-1 is the vector reflected ray, when ẞ is the incident ray

and a normal to the reflecting surface, § 105.

Interpretation of aßy when it is a vector, § 106.

Examples of variety in simple transformations, § 107.

Introduction to spherical trigonometry, §§ 108-113.

Representation, graphic, and by quaternions, of the spherical excess,
§§ 114, 115.

Loci represented by different equations-points, lines, surfaces, and solids,
§§ 116-119.

where f is linear and homogeneous in dq; but we cannot generally

Definition of the differential of a function of more quaternions than one.

d (gr)=qdr+dq.r, but not generally d (qr)=qdr+rdq. § 132.

Successive differentiation; Taylor's theorem, §§ 134, 135.

If the equation of a surface be

CHAPTER V.-THE SOLUTION OF EQUATIONS OF THE FIRST DEGREE.

97-128

The most general equation of the first degree in an unknown quaternion q,

may be written

ZV.aqb+S.cq=d,

where a, b, c, d are given quaternions. Elimination of Sq, and reduc-

tion to the vector equation

General proof that p3p is expressible as a linear function of p, p, and p3p,

§ 140.

Value of for an ellipsoid, employed to illustrate the general case,

§§ 141-143.

When is its own conjugate, the roots of the cubic are real; and the

is satisfied by a set of three real and mutually perpendicular vectors.

Geometrical interpretation of these results, §§ 162–166.

where a, B, y are any rectangular unit vectors whatever, we have

Sq=-mq, Vq=e. §§ 170-174.

Degrees of indeterminateness of the solution of a quaternion equation-

Examples, §§ 175-179.

The linear function of a quaternion is given by a symbolical biquadratic,

§ 180.

Particular forms of linear equations, §§ 181-183.

A quaternion equation of the mth degree in general involves a scalar
equation of degree m1, § 184.

Solution of the equation

q2=qa+b, § 185.

EXAMPLES TO CHAPTER V.

128-130

CHAPTER VI.-GEOMETRY OF THE STRAIGHT LINE AND PLANE.

CHAPTER VII.-THE SPHERE AND CYCLIC CONE. .

151-167

EXAMPLES TO CHAPTER VII.

.

167-170

CHAPTER VIII.-SURFACES OF THE SECOND ORDER. . 171-191

EXAMPLES TO CHAPTER VIII.

с

192-196

CHAPTER IX.-GEOMETRY OF CURVES AND SURFACES. 197-238