CHAP. I.]

CONSTRUCTION OF THE SERIES.

359

the feet of the perpendiculars let fall on these three sides from the points A, B, C; also let м and N be two auxiliary points, determined by the equations,

and let B", c" be two other auxiliary points, on the sides b and c, or on those sides prolonged, which satisfy these two other equations,

(2.) Then the perpendiculars to these last sides, CA and AB, erected at these last points, B" and c", will intersect each other in the point D, which completes (305) the spherical parallelogram BACD; and the foot of the perpendicular from this point D, on the third side BC of the given triangle, will coincide (comp. 305, (2.)) with the foot D' of the perpendicular on the same side from N; so that this last perpendicular ND' is one locus of the point D.

(3.) To obtain another locus for that point, adapted to our present purpose, let E denote now that new point in which the two diagonals, AD and BC, intersect each other; then because (comp. 297, (2.)) we have the expression,

VI... OE = u(mẞ+ny), whence VII... sin BE: sin EC = n; m = COS BA': cos a'c; the diagonal AD thus dividing the arc BC into segments, of which the sines are proportional to the cosines of the adjacent sides of the given triangle, or to the cosines of their projections BA' and A'C on BC; so that the greater segment is adjacent to the lesser side, and the middle point M of BC (1.) lies between the points a' and E.

(4.) The intersection E is therefore a known point, and the great circle through A and E is a second known locus for

D; which point may therefore be found, as the intersection of the arc AE prolonged, with the perpendicular ND' from N (1.). And because E lies (3.) beyond the middle point M of BC, with respect to the foot A' of the perpendicular on BC from A, but (as it is easy to prove) not so far beyond M as the point D', or in other words falls between M and D' (when the arc BC is, as above supposed, less than a quadrant), the prolonged arc AE cuts ND' between N and D'; or in other words, the perpendicular distance of the sought fourth point D, from the given diagonal BC of the parallelogram, is less than the distance of the

given second point A, from the same given diagonal. (Compare the annexed Fig. 73.)

It will be observed that M, N, E have not here the same significations as in

(5.) Proceeding next (305) to derive a new point A from B, D, C, as D has been derived from B, A, C, we see that we have only to determine a new* auxiliary point F, by the equation,

VIII. . . EM = ~ MF;

and then to draw DF, and prolong it till it meets AA' in the required point A1, which will thus complete the second parallelogram, BDCA1, with BC (as before) for a given diagonal.

(6.) In like manner, to complete (comp. 305, (5.)), the third parallelogram, BAICDI, with the same given diagonal BC, we have only to draw the arc AE, and prolong it till it cuts ND' in D1; after which we should find the point A of a fourth successive parallelogram BD¡CA2, by drawing D1F, and so on for ever.

(7.) The constant and indefinite tendency, of the derived points D, D1, . . to the limit-point D', and of the other (or alternate) derived points A1, A2, to the other limit-point a', becomes therefore evident from this new construction; the final (or limiting) results of which, we may express by these two equations (comp. again 305, (5.)),

IX. . . D ̧ = D' ; A» = A'.

(8.) But the smallness (305) of the first deviation AA1, when the sides of the given triangle ABC are small, becomes at the same time evident, by means of the same construction, with the help of the formula VII.; which shows that the intervalt EM, or the equal interval MF (5.), is small of the third order, when the sides of the given triangle are supposed to be small of the first order: agreeing thus with the equation 305, VIII.

(9.) The theory of such spherical parallelograms admits of some interesting applications, especially in connexion with spherical conics; on which however we cannot enter here, beyond the mere enunciation of a Theorem, of which (comp. 271) the proof by quaternions is easy :

Fig. 68; and that the present letters c' and d'' correspond to Q and R in that Figure.

* This new point, and the intersection of the perpendiculars of the given triangle, are evidently not the same in the new Figure 73, as the points denoted by the same letters, F and P, in the former Figure 68; although the four points A, B, C, D are conceived to bear to each other the same relations in the two Figures, and indeed in Fig. 67 also; BACD being, in that Figure also, what we have proposed to call a spherical parallelogram. Compare the Note to (3.).

The formula VII. gives easily the relation,

hence the interval EM is small of the third order, in the case (8.) here supposed; and

π

generally, if a<, as in (1.), while b and c are unequal, the formula shows that this interval EM is less than MA', or than D'M, so that E falls between м and D', as in (4.). This Theorem was cominunicated to the Royal Irish Academy in June, 1845, as a consequence of the principles of Quaternions. See the Proceedings of that date (Vol. III., page 109).

CHAP. I.] THIRD INTERPRETATION OF A PRODUCT.

361

"If KLMN be any spherical quadrilateral, and.1 any point on the sphere; if also we complete the spherical parallelograms,

and determine the poles E and F of the diagonals KM and LN of the quadrilateral : then these two poles are the foci* of a spherical conic, inscribed in the derived quadrilateral ABCD, or touching its four sides."

(10.) Hence, in a notation+ elsewhere proposed, we shall have, under these conditions of construction, the formula:

(11.) Before closing this Article and Section, it seems not irrelevant to remark, that the projection y' of the unit-vector y, on the plane of a and ẞ, is given by the formula,

and that therefore the point P, in which (see again Fig. 73) the three arcual perpendiculars of the triangle ABC intersect, is on the vector,

XIII... pa tan A+ ẞ tan B + y tan c.

(12.) It may be added, as regards the construction in 305, (2.), that the right lines,

XIV... PP1, P1P2, P2P3, P3P4,...

however far their series may be continued, intersect the given plane BOC, alternately, in two points s and T, of which the vectors are,

and which thus become two fixed points in the plane, when the position of the point P in space is given, or assumed.

SECTION 9.-On a Third Method of interpreting a Product or Function of Vectors as a Quaternion; and on the Consistency of the Results of the Interpretation so obtained, with those which have been deduced from the two preceding Methods of the present Book.

307. The Conception of the Fourth Proportional to Three Rectangular Unit-Lines, as being itself a species of Fourth Unit in Geometry, is eminently characteristic of the present Calcu lus; and offers a Third Method of interpreting a Product of two Vectors as a Quaternion: which is however found to be

* In the language of modern geometry, the conic in question may be said to touch eight given arcs; four real, namely the sides AB, BC, CD, DA; and four ima ginary, namely two from each of the focal points, E and F.

+ Compare the Second Note to page 295.

consistent, in all its results, with the two former methods (278, 284) of the present Book; and admits of being easily extended to products of three or more lines in space, and generally to Functions of Vectors (289). In fact we have only to conceive*

*It was in a somewhat analogous way that Des Cartes showed, in his Geometria (Schooten's Edition, Amsterdam, 1659), that all products and powers of lines, considered relatively to their lengths alone, and without any reference to their directions, could be interpreted as lines, by the suitable introduction of a line taken for unity, however high the dimension of the product or power might be. Thus (at page 3 of the cited work) the following remark occurs:—

"Ubi notandum est, quòd per a2 vel b3, similésve, communiter, non nisi lineas omnino simplices concipiam, licèt illas, ut nominibus in Algebra usitatis utar, Quadrata aut Cubos, &c. appellem."

But it was much more difficult to accomplish the corresponding multiplication of directed lines in space; on account of the non-existence of any such line, which is symmetrically related to all other lines, or common to all possible planes (comp. the Note to page 248). The Unit of Vector-Multiplication cannot properly be itself a Vector, if the conception of the Symmetry of Space is to be retained, and duly combined with the other elements of the question. This difficulty however disappears, at least in theory, when we come to consider that new Unit, of a scalar kind (300), which has been above denoted by the temporary symbol u, and has been obtained, in the foregoing Section, as a certain Fourth Proportional to Three Rectangular Unit-Lines, such as the three co-initial edges, Ab, ac, ad of what we have called an Unit-Cube for this fourth proportional, by the proposed conception of it, undergoes no change, when the cube ABCD is in any manner moved, or turned; and therefore may be considered to be symmetrically related to all directions of lines in space, or to all possible vections (or translations) of a point, or body. In fact, we conceive its determination, and the distinction of it (as +u) from the opposite unit of the same kind (u), to depend only on the usual assumption of an unit of length, combined with the selection of a hand (as, for example, the right hand), rotation towards which hand shall be considered to be positive, and contrasted (as such) with rotation towards the other hand, round the same arbitrary axis. Now in whatever manner the supposed cube may be thrown about in space, the conceived rotation round the edge AB, from AC to AD, will have the same character, as right-handed or left-handed, at the end as at the beginning of the motion. If then the fourth proportional to these three edges, taken in this order, be denoted by +u, or simply by + 1, at one stage of that arbitrary motion, it may (on the plan here considered) be denoted by the same symbol, at every other stage: while the opposite character of the (conceived) rotation, round the same edge AB, from AD to AC, leads us to regard the fourth proportional to AB, AD, AC as being on the contrary equal to -u, or to 1. It is true that this conception of a new unit for space, symmetrically related (as above) to all linear directions therein, may appear somewhat abstract and metaphysical; but readers who think it such can of course confine their attention to the rules of calculation, which have been above derived from it, and from other connected considerations: and which have (it is hoped) been stated and exemplified, in this and in a former Volume, with sufficient clearness and fullness.

CHAP. I.]

CONCEPTION OF THE FOURTH UNIT.

363

that each proposed vector, a, is divided by the new or fourth unit, u, above alluded to; and that the quotient so obtained, which is always (by 303, VIII.) the right quaternion I ̈1a, whereof the vector a is the index, is substituted for that vector: the resulting quaternion being finally, if we think it convenient, multiplied into the same fourth unit. For in this way we shall merely reproduce the process of 284, or 289, although now as a consequence of a different train of thought, or of a distinct but Consistent Interpretation: which thus conducts, by a new Method, to the same Rules of Calculation as before.

(1.) The equation of the unit-sphere, p2 + 1 = 0 (282, XIV.), may thus be conceived to be an abridgment of the following fuller equation:

the quotient p: u being considered as equal (by 303) to the right quaternion, I ́1p, which must here be a right versor (154), because its square is negative unity. (2.) The equation of the ellipsoid,

may be supposed, in like manner, to be abridged from this other equation:

(3.) We might also write these equations, of the sphere and ellipsoid, under these other, but connected forms:

with intepretations which easily offer themselves, on the principles of the foregoing Section.

(4.) It is, however, to be distinctly understood, that we do not propose to adopt this Form of Notation, in the practice of the present Calculus: and that we merely suggest it, in passing, as one which may serve to throw some additional light on the Conception, introduced in this Third Book, of a Product of two Vectors as a Quaternion.

(5.) In general, the Notation of Products, which has been employed throughout the greater part of the present Book and Chapter, appears to be much more convenient, for actual use in calculation, than any Notation of Quotients: either such as has been just now suggested for the sake of illustration, or such as was employed in the Second Book, in connexion with that First Conception of a Quaternion (112), to which that Book mainly related, as the Quotient of two Vectors (or of two directed lines in space). The notations of the two Books are, however, intimately connected, and the former was judged to be an useful preparation for the latter, even as