corner of the triangle. Let t and t, determine the others. Then, if w1, w1, represent the vectors of the points of inter

3

section of the tangents with the sides, we easily find

or

Hence, by § 30, the proposition is proved.

(.) Other interesting examples of this method of treating curves will, of course, suggest themselves to the student. Thus

p = a cost+ß sin t

p = ax +ß√1 — x2

represents an ellipse, of which the given vectors a and B are semi-conjugate diameters.

Again,

B

p = at +

or p = a tan x+ẞ cot a

t

evidently represents a hyperbola referred to its asymptotes.

But, so far as we have yet gone, as we are not prepared to determine the lengths or inclinations of vectors, we can only investigate a very small class of the properties of curves, represented by such equations as those above written.

(7.) We may now, in extension of the statement in § 29,

make the obvious remark that

p = Σpa

is the equation of a curve in space, if the numbers P1, P1, &c. are functions of one indeterminate. In such a case the equation

But, if P1, P2, &c. be functions of two indeterminates, the locus of the extremity of p is a surface; whose equation is sometimes written p = $(t, u).

belongs to a central surface of the second order, of which a, ß, y are the directions of conjugate diameters. If a, b, c be all positive, the surface is an ellipsoid.

32. In Example (f) above we performed an operation equivalent to the differentiation of a vector with reference to a single numerical variable of which it was given as an explicit function. As this process is of very great use, especially in quaternion investigations connected with the motion of a particle or point; and as it will afford us an opportunity of making a preliminary step towards overcoming the novel difficulties which arise in quaternion differentiation; we will devote a few sections to a more careful exposition of it.

33. It is a striking circumstance, when we consider the way in which Newton's original methods in the Differential Calculus have been decried, to find that Hamilton was obliged to employ them, and not the more modern forms, in order to overcome the characteristic difficulties of quaternion differentiation. Such a thing as a differential coefficient has absolutely no meaning in

quaternions, except in those special cases in which we are dealing with degraded quaternions, such as numbers, Cartesian cöordinates, &c. But a quaternion expression has always a differential, which is, simply, what Newton called a fluxion.

As with the Laws of Motion, the basis of Dynamics, so with the foundations of the Differential Calculus; we are gradually coming to the conclusion that Newton's system is the best after all.

34. Suppose p to be the vector of a curve in space. Then, generally, p may be expressed as the sum of a number of terms, each of which is a multiple of a given vector by a function of some one indeterminate; or, as in § 31 (7), if P be a point on the curve, OP = p = $(t).

And, similarly, if Q be any other point on the curve,

where it is any

OQ=P1 = $(t1) = $(t+dt),

number whatever.

The vector-chord PQ is therefore, rigorously,

δρ = ρ1 ρ = φ (t+δt)φέ.

35. It is obvious that, in the present case, because the vectors involved in & are constant, and their numerical multipliers alone vary, the expression (t+dt) is, by Taylor's Theorem, equivalent to

And we are thus entitled to write, when it has been made in

In such a case as this, then, we are permitted to differentiate,

or to form the differential coefficient of, a vector according to the ordinary rules of the Differential Calculus. But great additional insight into the process is gained by applying Newton's method.

that the value of t for the vector of point P of the curve denotes the interval which has elapsed (since a fixed epoch) when the moving point has reached the extremity of that vector. If, then, dt represent any interval, finite or not, we see that

will be the vector of the point after the additional interval dt.

2

But this, in general, gives us little or no information as to the velocity of the point at P. We shall get a better approximation by halving the interval dt, and finding Q., where OQ2 = $(t + 1⁄2 dt), as the position of the moving point at that time. Here the vector virtually described in dt is PQ. To find, on this supposition, the vector described in dt, we must double this, and we find, as a second approximation to the vector which the moving point would have described in time dt, if it had moved for that period in the direction and with the velocity it had at P,

Pq2=2PQ2 = 2(OQ,-OP)

= 2 { p (t + } dt) —$t}.

And so on, each step evidently leading us nearer the sought truth. Hence, to find the vector which would have been described in time dt had the circumstances of the motion at P remained undisturbed, we must find the value of

dp = Pq= Lx=xx {p (t+= dt)—pt}.

We have seen that in this particular case we may use Taylor's Theorem. We have, therefore,

37. But it is to be most particularly remarked that in the whole of this investigation no regard whatever has been paid to the magnitude of dt. The question which we have now answered may be put in the form-A point describes a given curve in a given manner. At any point of its path its motion suddenly ceases to be accelerated. What space will it describe in a definite interval? As Hamilton well observes, this is, for a planet or comet, the case of a "celestial Attwood's machine."

38. If we suppose the variable, in terms of which p is expressed, to be the arc, s, of the curve measured from some fixed point, we find as before

dp = '(t)dt =

$'(t)ds

ds

dt