which the plane (119), and the axis (127), coincide with those of q, several general consequences easily follow. Thus we have generally, by principles already established, the relations: I... 4 Vq π 2 ; II. . . Ax. Vq = Ax. UVq = Ax. q ; III... KVq = - Vq, or KV-V (144); and therefore, = · (Vg)2 = − (TVg)2 = - NVq,* because, by the general decomposition (188) of a quaternion into factors, we have VII... Vq=TVq. UVq. We have also (comp. 196, VI.), and VIII. . . VSq = 0, or VS=0 (202, VII.); = IX... VVq=Vq, or V2- VV = V (202, IX.); because conjugate quaternions have opposite right parts, by the definitions in 137, 202, and by the construction of Fig. 36. For the same reason, we have this other general formula, but we had XI... Kq = Sq - Vq, or K=S-V; q = Sq + Vq, or 1=S+ V, by 202, III., IV.; hence not only, by addition, q+Kq = 2Sq, or 1 + K = 2S, as in 196, I., but also, by subtraction, XII... 9- Kq=2Vq, or 1-K=2V; whence the Characteristic, V, of the Operation of taking the Right Part of a Quaternion (comp. 132, (6.); 137; 156; 187; 196), may be defined by either of the two following symbolical equations: XIII. . . V = 1-S (202, IV.); XIV. . . V = ¦ (1 − K) ; whereof the former connects it with the characteristic S, and Compare the Note to page 130. CHAP. 1.] PROPERTIES OF THE RIGHT (OR VECTOR) PART. 195 the latter with the characteristic K; while the dependence of K on S and V is expressed by the recent formula XI.; and that of S on K by 196, II'. Again, if the line oв, in Fig. 50, be multiplied (15) by any scalar coefficient, the perpendicular BB' is evidently multiplied by the same; hence, generally, XV... Vxq = xVq, if x be any scalar; XVI... Vq= Tq.VUq, and XVII... TVq = Tq.TVUq. But the consideration of the right-angled triangle, OB'в, in the same Figure, shows that we arrive then thus at the following general and useful formula, connecting quaternions with trigonometry anew: XIX. . . TVUq = sin ≤ q ; by combining which with the formula, SUq=cos q (196, XVI.), we arrive at the general relation : XX. . . (SUq)2 + (TVUq)2 = 1 ; which may also (by XVII., and by 196, IX.) be written thus: XXI. . . (Sq)2 + (TVq)2 = (Tq)2 ; and might have been immediately deduced, without sines and cosines, from the right-angled triangle, by the property of the square of the hypotenuse, under the form, (T.OB')2 + (T.B'B)2 = (T.OB)2. The same important relation may be expressed in various other ways; for example, we may write, XXII... Nq = Tq2 = Sq2 – Vq2, where it is assumed, as an abridgment of notation (comp. 199, VII., VIII.), that XXIII. . . Vq2 = (Vg), but that XXIV... V. q2 = V (q2), the import of this last symbol remaining to be examined. And because, by the definition of a norm, and by the properties of Sq and Vq, XXV... NSq = Sq, but XXVI. . . NVg = − Vq2, we may write also, XXVII. . . Nq = N(Sq + Vq) = NSq+ NVq ; a result which is indeed included in the formula 200, VIII., since that equation gives, generally, XXVIII. . . N(q+x) = Nq + Nx, if q = π x being, as usual, any scalar. It may be added that because (by 106, 143) we have, as in algebra, the identity, XXIX... - (q' + q) = − q' − q› the opposite of the sum of any two quaternions being thus equal to the sum of the opposites, we may (by XI.) establish this other general formula: XXX... - Kq=Vq-Sq; the opposite of the conjugate of any quaternion q having thus the same right part as that quaternion, but an opposite scalar part. (1.) From the last formula it may be inferred, that if qKq, then Vq'=+Vq, but Sq'-Sq; which two last relations might have been deduced from 138 and 143, without the introduction of the characteristics S and V. like the equation of 203, (5.), expresses that the locus of P is the right cylinder, or cylinder of revolution, with oa for its axis, which passes through the point B. (3.) The system of the two equations, like the corresponding system in 203, (6.), represents generally an elliptic section of the same right cylinder; but if it happen that y a, the section then becomes circular. represents the circle,* in which the cylinder of revolution, with oa for axis, and with (1-2) Ta for radius, is perpendicularly cut by a plane at a distance = + Ta from o; the vector of the centre of this circular section being xa. (5.) While the scalar x increases (algebraically) from 1 to 0, and thence to + 1, the connected scalar √(1 − x2) at first increases from 0 to 1, and then decreases from 1 to 0; the radius of the circle (4.) at the same time enlarging from zero to a maximum Ta, and then again diminishing to zero; while the position of the centre of the circle varies continuously, in one constant direction, from a first limit-point a', if OA'=-a, to the point A, as a second limit. = (6.) The locus of all such circles is the sphere, with AA' for a diameter, and therefore with o for centre; namely, the sphere which has already been represented by the equation Tp = Ta of 186, (2.); or by T=1, of 187, (1.); or by a = 0, of 200, (11.); obtained by eliminating x between the two recent equations (4). (7.) It is easy, however, to return from the last form to the second, and thence to the first, or to the third, by rules of calculation already established, or by the general relations between the symbols used. In fact, the last equation (6.) may be written, by XXII., under the form, (8.) Conversely, the sphere through A, with o for centre, might already have been seen, by the first definition and property of a norm, stated in 145, (11.), to ad ρ a 1 ; and there mit (comp. 145, (12.)) of being represented by the equation N fore, by XXII., under the recent form (6.); in which if we write z to denote the variable scalar S, as in the first of the two equations (4.), we recover the second of those equations: and thus might be led to consider, as in (6.), the sphere in question By the word "circle," in these pages, is usually meant a circumference, and not an area; and in like manner, the words "sphere," "cylinder," " cone," &c., are usually here employed to denote surfaces, and not volumes. as the locus of a variable circle, which is (as above) the intersection of a variable cylinder, with a variable plane perpendicular to its axis. (9.) The same sphere may also, by XXVII., have its equation written thus, N(S+VC) -1; or T(S+V)-1 a a a α 1. (10.) If, in each variable plane represented by the first equation (4.), we conceive the radius of the circle, or that of the variable cylinder, to be multiplied by any constant and positive scalar a, the centre of the circle and the axis of the cylinder remaining unchanged, we shall pass thus to a new system of circles, represented by this new system of equations, (11.) The locus of these new circles will evidently be a Spheroid of Revolution; the centre of this new surface being the centre o, and the axis of the same surface being the diameter AA', of the sphere lately considered: which sphere is therefore either inscribed or circumscribed to the spheroid, according as the constant a> or <1; because the radii of the new circles are in the first case greater, but in the second case less, than the radii of the old circles; or because the radius of the equator of the spheroid = aTa, while the radius of the sphere - Ta. (12.) The equations of the two co-axal cylinders of revolution, which envelope respectively the sphere and spheroid (or are circumscribed thereto) are: represents (comp. (3.)) a variable ellipse, if the scalar ≈ be still treated as a va riable. (14.) The result of the elimination of a between the two last equations, namely this new equation, |