Hence the polar of any point is parallel to the tangent at the extremity of the diameter on which the point lies, and its intersection with that diameter is as far beyond the vertex as the pole is within, and vice versá. (j.) As another example let us prove the following theorem. If a triangle be inscribed in a parabola, the three points in which the sides are met by tangents at the angles lie in a straight line. Since O is any point of the curve, we may take it as one corner of the triangle. Let t and t1 determine the others. Then, if 1, 2, 3 represent the vectors of the points of intersection of the tangents with the sides, we easily find or Hence, by § 30, the proposition is proved. (k.) Other interesting examples of this method of treating curves will, of course, suggest themselves to the student. Thus p = a cost + B sin t p = ax+ẞ√1-22 represents an ellipse, of which the given vectors a and ẞ are semiconjugate diameters. Again, p = at +/ t or p = a tan x+ß cotx . evidently represents a hyperbola referred to its asymptotes. But, so far as we have yet gone with the explanation of the calculus, as we are not prepared to determine the lengths or inclinations of vectors, we can investigate only 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 ρ = Σμα is the equation of a curve in space, if the numbers P1, P2, &c. are functions of one indeterminate. In such a case the equation is sometimes written p = $(t). 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, B, y are the directions of conjugate diameters. If a, b, c be all positive, the surface is an ellipsoid. 32.] In Example (ƒ) 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. From the very nature of the question it is obvious that the length of dp must in this case be ds. This remark is of importance, as we shall see later; and it may therefore be useful to obtain afresh the above result without any reference to time or velocity. 39.] Following strictly the process of Newton's VIIth Lemma, let us describe on Pq2 an arc similar to PQ2, and so on. Then obviously, as the subdivision of ds is carried farther, the new arc (whose length is always ds) more and more nearly coincides with the line which expresses the corresponding approximation to dp. 40.] As a final example let us take the hyperbola This shews that the tangent is parallel to the vector In words, if the vector (from the centre) of a point in a hyperbola be one diagonal of a parallelogram, two of whose sides coincide with the asymptotes, the other diagonal is parallel to the tangent at the point. 41.] Let us reverse this question, and seek the envelope of a line which cuts off from two fixed axes a triangle of constant area. If the axes be in the directions of a and ẞ, the intercepts may evidently be written at and. Hence the equation of the line is (§ 30) p = at + x (−at). The condition of envelopment is, obviously, (see Chap. IX.) * We are not here to equate to zero the coefficients of dt and dx; for we must remember that this equation is of the form 0 = pa+qß, where Ρ and are numbers; and that, so long as a and B are actual and non-parallel vectors, the existence of such an equation requires p = 0, 9 = 0. From these, at once, a = 1, since da and dt are indeterminate. Thus the equation of the envelope is the hyperbola as before; a, ß being portions of its asymptotes. 42.] It may assist the student to a thorough comprehension of the above process, if we put it in a slightly different form. Thus the equation of the enveloping line may be written which gives p = at (1−x)+ß2, dp = 0 = ad (t (1 − x)) +ßd (2) · Hence, as a is not parallel to B, we must have d (t (1 − x)) = 0, d (2) = =0; and these are, when expanded, the equations we obtained in the preceding section. 43.] For farther illustration we give a solution not directly employing the differential calculus. The equations of any two of the enveloping lines are t and t1 being given, while x and x1 are indeterminate. or the intersection lies in the diagonal of the parallelogram on a, ß. COR. (2). If t1 = mt, where m is constant, Hence the locus of a point which divides in a given ratio a line cutting off a given area from two fixed axes, is a hyperbola of which these axes are the asymptotes. tt (t+t) constant the locus is a parabola; and so on. 44.] The reader who is fond of Anharmonic Ratios and Transversals will find in the early chapters of Hamilton's Elements of Quaternions an admirable application of the composition of vectors to these subjects. The Theory of Geometrical Nets, in a plane, and in space, is there very fully developed; and the method is shewn to include, as particular cases, the processes of Grassmann's Ausdehnungslehre and Möbius' Barycentrische Calcul. Some very curious investigations connected with curves and surfaces of the second and third orders are also there founded upon the composition of vectors. EXAMPLES TO CHAPTER I. 1. The lines which join, towards the same parts, the extremities of two equal and parallel lines are themselves equal and parallel. (Euclid, I. xxxiii.) 2. Find the vector of the middle point of the line which joins 1 T |