Page images
PDF
EPUB

CHAP. 1.] OTHER VIEW OF A FOURTH PROPORTIONAL.

349

Ba-ly = 1, as in the recent equations VI.; and similarly in other examples, of the class here considered.

(7.) The conclusion, that the Fourth Proportional to Three Rectangular Lines is a Scalar, may in several other ways be deduced, from the principles of the present Book. For example, with the recent suppositions, we may write,

VIII... ẞa-1=-Y,

YB-1=-a, ay-1=- ß;
VIII'... ya1=+ß, aß1=+y, By1=+a;

the three fourth proportionals VI. are therefore equal, respectively, to y2, - a2, - B2, and consequently to +1; while the corresponding expressions VI'. are equal to +ẞ2, + y2, + a2, and therefore to -1.

(8.) Or (comp. (3.)) we may write generally the transformation (comp. 282, XXI.*),

IX... Ba ̈1y = a. Bay, if a2=1: a2,

in which the factor a-2 is always a scalar, whatever vector a may be; while the vector part of the ternary product ẞay vanishes, by 294, III., when the recent conditions of rectangularity III. are satisfied.

(9.) Conversely, this ternary product Bay, and this fourth proportional Ba ̄1y, can never reduce themselves to scalars, unless the three vectors a, ẞ, y (supposed to be all actual (Art. 1)) are perpendicular each to each.

SECTION 8.-On an equivalent Interpretation of the Fourth Proportional to Three Diplanar Vectors, deduced from the Principles of the Second Book.

300. In the foregoing Section, we naturally employed the results of preceding Sections of the present Book, to assist ourselves in attaching a definite signification to the Fourth Proportional (297) to Three Diplanar Vectors; and thus, in order to interpret the symbol Bay, we availed ourselves of the interpretations previously obtained, in this Third Book, of a1 as a line, and of aß, aßy as quaternions. But it may be interesting, and not uninstructive, to inquire how the equivalent symbol,

[blocks in formation]

might have been interpreted, on the principles of the Second Book, without at first assuming as known, or even seeking to discover, any interpretation of the three lately mentioned symbols,

II... a, aẞ, aẞy.

It will be found that the inquiry conducts to an expression of the form,

* The formula here referred to should have been printed as Ra = 1: a = a1.

III... (ẞ: a). y=d+eu ;

where is the same vector, and e is the same scalar, as in the recent sub-articles to 297; while u is employed as a temporary symbol, to denote a certain Fourth Proportional to Three Rectangular Unit Lines, namely, to the three lines oq, OL', and or in Fig. 68; so that, with reference to the construction represented by that Figure, we should be led, by the principles of the Second Book, to write the equation:

IV... (OB: OA). Oc=OD.cos + (OL': OQ).OP.sin

And when we proceed to consider what signification should be attached, on the principles of the same Second Book, to that particular fourth proportional, which is here the coefficient of sin, and has been provisionally denoted by u, we find that although it may be regarded as being in one sense a Line, or at least homogeneous with a line, yet it must not be equated to any Vector: being rather analogous, in Geometry, to the Scalar Unit of Algebra, so that it may be naturally and conveniently denoted by the usual symbol 1, or + 1, or be equated to Positive Unity. But when we thus write u=1, the last term of the formula III. or IV., of the present Article, becomes simply e, or sin ; and while this term (or part) of the result comes to be considered as a species of Geometrical Scalar, the complete Expression for the General Fourth Proportional to Three Diplanar Vectors takes the Form of a Geometrical Quaternion: and thus the formula 297, XLVII., or 298, VIII., is reproduced, at least if we substitute in it, for the present, (B: a). y for Bay, to avoid the necessity of interpreting here the recent symbols II.

(1.) The construction of Fig. 68 being retained, but no principles peculiar to the Third Book being employed, we may write, with the same significations of c, p, &c., as before,

V... OB: OA = OR: OQ = cos c+ (OL': OQ) sin c;

VI... oc=oq. cos p + OP. sin p.

(2.) Admitting then, as is natural, for the purposes of the sought interpretation, that distributive property which has been proved (212) to hold good for the multiplication of quaternions (as it does for multiplication in algebra); and writing for abridgment,

VII... u = (OL': OQ). OP;

we have the quadrinomial expression :

VIII... (OB: OA). OC = OL'. sin c cos p+oQ. cos c cos p

+ OP. cos c sin p + u. sin c sin p;

in which it may be observed that the sum of the squares of the four coefficients of the

CHAP. I.]

SCALAR UNIT IN SPACE.

351

three rectangular unit-vectors, oq, OL', OP, and of their fourth proportional, u, is equal to unity.

(3.) But the coefficient of this fourth proportional, which may be regarded as a species of fourth unit, is

IX... sin c sin p = sin MN sin 2 = e;

we must therefore expect to find that the three other coefficients in VIII., when divided by cos 2, or by r, give quotients which are the cosines of the arcual distances of some point x upon the unit-sphere, from the three points L', Q, P; or that a point x can be assigned, for which

X... sin c cosp=r cos L'X; cos c cos p r cos QX ; cos e sin pr cos PX. (4.) Accordingly it is found that these three last equations are satisfied, when we substitute D for x; and therefore that we have the transformation,

XI... OL'. sin c cos p + oq. cos c cosp+OP. cos e sin p = on. cos2d, whence follow the equations IV. and III.; and it only remains to study and interpret the fourth unit, u, which enters as a factor into the remaining part of the quadrinomial expression VIII, without employing any principles except those of the Second Book: and therefore without using the Interpretations 278, 284, of Ba, &c.

301. In general, when two sets of three vectors, a, ß, y, and a', B', ', are connected by the relation,

[blocks in formation]

and to say that these two fourth proportionals (297), to a, ß, 7, and to a', B', y', are equal to each other: whatever the full signification of each of these two last symbols III., supposed for the moment to be not yet fully known, may be afterwards found to be. In short, we may propose to make it a condition of the sought Interpretation, on the principles of the Second Book, of the phrase,

"Fourth Proportional to Three Vectors,"

and of either of the two equivalent Symbols 300, I., that the recent Equation III. shall follow from I. or II.; just as, at the commencement of that Second Book, and before concluding (112) that the general Geometric Quotient B: a of any two lines in space is a Quaternion, we made it a condition (103) of the interpretation of such a quotient, that the equation (B: a). a = ẞ should be satisfied.

302. There are however two tests (comp. 287), to which the recent equation III. must be submitted, before its final adoption; in

order that we may be sure of its consistency, Ist, with the previous interpretation (226) of a Fourth Proportional to Three Complanar Vectors, as a Line in their common plane; and IInd, with the general principle of all mathematical language (105), that things equal to the same thing, are to be considered as equal to each other. And it is found, on trial, that both these tests are borne: so that they form no objection to our adopting the equation 301, III., as true by definition, whenever the preceding equation II., or I., is satisfied.

(1.) It may happen that the first member of that equation III. is equal to a line , as in 226; namely, when a, ẞ, y are complanar. In this case, we have by II. the equation,

δ δ γ B'

[blocks in formation]

γ

=

[blocks in formation]

"

[ocr errors]
[ocr errors]

a

Y;

so that a', B', y' are also complanar (among themselves), and the line d is their fourth proportional likewise: and the equation III. is satisfied, both members being symbols for one common line, d, which is in general situated in the intersection of the two planes, aßy and a'ß'y'; although those planes may happen to coincide, without disturbing the truth of the equation.

(2.) Again, for the more general case of diplanarity of a, 3, y, we may conceive that the equation* II. co-exists with this other of the same form,

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

if the definition 301 be adopted. If then that definition be consistent with general principles of equality, we ought to find, by III. and VI., that this third equation between two fourth proportionals holds good:

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

when the equations II. and V. are satisfied. And accordingly, those two equations give, by the general principles of the Second Book, respecting quaternions considered as quotients of vectors, the transformation,

[blocks in formation]

303. It is then permitted to interpret the equation 301, III., on the principles of the Second Book, as being simply a transformation (as it is in algebra) of the immediately preceding equation II., or I.; and therefore to write, generally,

I... q7=q'', if II... q(y: y) = q';

* In this and other cases of reference, the numeral cited is always supposed to be the one which (with the same number) has last occurred before, although perhaps it may have been in connexion with a shortly preceding Article. Compare 217, (1.).

CHAP. I.]

FOURTH PROPORTIONAL RESUMED.

353

where are any two vectors, and q, q' are any two quaternions, which satisfy this last condition. Now, if v and v' be any two right quaternions, we have (by 193, comp. 283) the equation,

[merged small][merged small][merged small][merged small][ocr errors]

It follows, then,

by the principle which has just been enunciated. that "if a right Line (Iv) be multiplied by the Reciprocal (v) of the Right Quaternion (v), of which it is the Index, the Product (v'Iv) is independent of the Length, and of the Direction, of the Line thus operated on;" or, in other words, that this Product has one common Value, for all possible Lines (a) in Space: which common or constant value may be regarded as a kind of new Geometrical Unit, and is equal to what we have lately denoted, in 300, III., and VII., by the temporary symbol u; because, in the last cited formula, the line op is the index of the right quotient oq: OL'. Retaining, then, for the moment, this symbol, u, we have, for every line a in space, considered as the index of a right quaternion, v, the four equations:

VI...

v ̄1a = u ;

VII... a = vu;
IX... v ̄' = u: a;

VIII. . . v = a: u;

in which it is understood that a = Iv, and the three last are here regarded as being merely transformations of the first, which is deduced and interpreted as above. And hence it is easy to infer, that for any given system of three rectangular lines a, ẞ, 7, we have the general expression:

X... (ẞ: a). y=xu, if «Ιβ, βι, για;

where the scalar co-efficient, x, of the new unit, u, is determined by the equation,

=

XI... x (TB: Ta). Ty, according as XII... Uy=+ Ax. (a: B). This coefficient x is therefore always equal, in magnitude (or absolute quantity), to the fourth proportional to the lengths of the three given lines aẞy; but it is positively or negatively taken, according as the rotation round the third line y, from the second line ẞ, to the first line a, is itself positive or negative: or in other words, according as the rotation round the first line, from the second to the third, is on the contrary negative or positive (compare 294, (3.) ).

(1.) In illustration of the constancy of that fourth proportional which has been, for the present, denoted by u, while the system of the three rectangular unit-lines

« PreviousContinue »