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The condition that p, a, ß may terminate in the same line is
Pp+qa+roß = 0,
P+q + r = 0.
is the condition that the extremities of four vectors lie in one plane. § 30.
where s is the length of the arc, dp is a vector tangent to the curve, and
its length is ds. SS 38, 39.
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 neither parallel nor perpendicular the quotient in general
involves four distinct numbers—and is thus a QUATERNION. § 47.
It is thus decomposable into two factors, whose order is indifferent, the
are denoted by Tq, and Uq. § 48.
U.q-1 = (U9)-1 $ 51,
and qKq = Ką.9 = (T9)2. $ 52.
K(gr) = Kr. Kq. $ 55.
Proof of the Associative Law of Multiplication
p.qr = Pq.1. SS 57-60.
[Digression on Spherical Conics. § 59*.]
Composition of quadrantal versors in planes at right angles to each other.
Calling them i, j, k, we have
The product, and the quotient, of two vectors at right angles to each other is
a third perpendicular to both. Hence
Quadrinomial expression for a quaternion
V. Vaß Vyd = al.Byd - BS.ard,
= δS. αβγ- γ. αβο,
that of perpendicular vectors, a vector; if quaternions are to deal with
space indifferently in all directions. 893.
EXAMPLES TO CHAPTER II.
Definition of the differential of a function of more quaternions than one.
Value of ø for an ellipsoid, employed to illustrate the general theory.
$ $ 141-143.
the functions and ' are said to be conjugate, and
m φ-1Vλμ = Vφλφ ́μ.
(which, like m, are Invariants,)
X = m2-0,
mp-1 = m,-m20 +02. $$ 147, 148.
is satisfied by a set of three real and mutually perpendicular vectors.
Geometrical interpretation of these results. $$ 162–166.
pp = fp+hV. (i + ek) piek)
Another transformation is
aaVap + BBSBp. $$ 167-169.
Sp($+g) –'p = 0, and Sp (p+h)->p = 0
Proof that when p is not self-conjugate
φρ = φρ + Vερ.
where a, b, y are any rectangular unit-vectors whatever, we have
Sq= -ma, Vq = 6.
CHAPTER VIII.-SURFACES OF THE SECOND ORDER .
EXAMPLES TO CHAPTER VIII.