CHAP. I.] SECOND INTERPRETAtion of a product.

309

we should only have had to conceive (as we always may), that the two proposed factors, a and ß, are the indices of two right quaternions, v and v', and to multiply these latter, in the same order. For thus we should have been led to establish the formula,

are understood to denote the two right quaternions, whereof the two lines a and ẞ are the indices.

(1.) To establish now the substantial identity of these two interpretations, 278 and 284, of a binary product of vectors ẞa, notwithstanding the difference of form of the definitional equations by which they have been expressed, we have only to observe that it has been found, as a theorem (194), that

IV... v'v Iv': I (1 : v) = Iv': IRe;

but the definition (258) of Ra gave us the lately cited equation, RIv=IRv; we have therefore, by the recent formula II., the equation,

V... Iv'. Iv Iv': RIv; or VI... ẞ.a = ß: Ra,

as in 278, I.; a and ẞ still denoting any two vectors. The two interpretations therefore coincide, at least in their results, although they have been obtained by different processes, or suggestions, and are expressed by two different formula.

(2.) The result 279, II., respecting conjugate products of vectors, corresponds thus to the result 191, (2.), or to the first formula of 223, (1.).

(3.) The two formulæ of 279, (1.) and (2.), respecting the scalar and right parts of the product ßa, answer to the two other formulæ of the same sub-article, 223, (1.), respecting the corresponding parts of v ́v.

(4.) The doubly distributive property (280), of vector-multiplication, is on this plan seen to be included in the corresponding but more general property (212), of multiplication of quaternions.

(5.) By changing IVq, IVq', t, t', and ô, to a, ẞ, a, b, and y, in those formula of Art. 208 which are previous to its sub-articles, we should obtain, with the recent definition (or interpretation) II. of Ba, several of the consequences lately given (in sub-arts. to 281), as resulting from the former definition, 278, I. Thus, the equations,

VI., VII., VIII,, IX., X., XI., XII., XXII., and XXIII., of 281, correspond to, and may (with our last definition) be deduced from, the formulæ,

V., VI., VIII., XI., XII., XXII., XX., XIV., and XVI., XVIII.,

of 208. (Some of the consequences from the sub-articles to 208 have been already considered, in 281, (11.))

(6.) The geometrical properties of the line IVßa, deduced from the first definition (278) of ẞa in 281, (3.) and (4.), (namely, the positive rotation round that line, from 3 to a; its perpendicularity to their plane; and the representation by the same line of the paralellogram under those two factors, regard being had to units of length and of area,) might also have been deduced from 223, (4.), by means of the second definition (284), of the same product, ẞa.

SECTION 4.-On the Symbolical Identification of a Right Quaternion with its own Index: and on the Construction of a Product of Two Rectangular Lines, by a Third Line, rectangular to both.

285. It has been seen, then, that the recent formula 284, II. or III., may replace the formula 278, I., as a second definition of a product of two vectors, which conducts to the same consequences, and therefore ultimately to the same interpretation of such a product, as the first. Now, in the second formula, we have interpreted that product, ẞa, by changing the two factor-lines, a and ẞ, to the two right quaternions, v and v', or Ia and I'ẞ, of which they are the indices; and by then defining that the sought product ẞa is equal to the product v'v, of those two right quaternions. It becomes, therefore, important to inquire, at this stage, how far such substitution, of I-1a for a, or of v for Iv, together with the converse substitution, is permitted in this Calculus, consistently with principles already established. For it is evident that if such substitutions can be shown to be generally legitimate, or allowable, we shall thereby be enabled to enlarge greatly the existing field of interpretation: and to treat, in all cases, Functions of Vectors, as being, at the same time, Functions of Right Quaternions.

286. We have first, by 133 (comp. 283, I.), the equality,
I... Iẞ I'a, if ẞ=a.

In the next place, by 206 (comp. 283, II.), we have the formula of addition or subtraction,

II... I1(ẞa) = I'ẞ+I1a;

with these more general results of the same kind (comp. 207 and 99),

III. . . Ι 'Σα = ΣΙ'α; IV... I'Exα = ΣxI ̈1a.

CHAP. I.] RIGHT QUATERNION EQUAL TO ITS INDEX.

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In the third place, by 193 (comp. 283, III.), we have, for division, the formula,

V... I'ẞ: I'a =ẞ: a;

while the second definition (284) of multiplication of vectors, which has been proved to be consistent with the first definition (278), has given us the analogous equation,

VI... Iẞ. I'1a = ẞ. a = ẞa.

It would seem, then, that we might at once proceed to define, for the purpose of interpreting any proposed Function of Vectors as a Quaterternion, that the following general Equation exists:

VII. . . I1a = a; or VIII. . . Iv = v, if

or still more briefly and symbolically, if it be understood that the subject of the operation I is always a right quaternion,

IX. I = 1.

But, before finally adopting this conclusion, there is a case (or rather a class of cases), which it is necessary to examine, in order to be certain that no contradiction to former results can ever be thereby caused.

287. The most general form of a vector-function, or of a vector regarded as a function of other vectors and of scalars, which was considered in the First Book, was the form (99, comp. 275),

I... p = Σxa;

and we have seen that if we change, in this form, each vector a to the corresponding right quaternion I'a, and then take the index of the new right quaternion which results, we shall thus be conducted to precisely the same vector p, as that which had been otherwise obtained before; or in symbols, that

II. . . Σxa = IΣxI-'a (comp. 286, IV.).

But another form of a vector-function has been considered in the Second Book; namely, the form,

in which a, ẞ, y, ô, ε... are any odd number of complanar vectors. And before we accept, as general, the equation VII. or VIII. or IX. of 286, we must inquire whether we are at liberty to write, under the same conditions of complanarity, and with the same signification of the vector p, the equation,

288. To examine this, let there be at first only three given complanar vectors, ||a, ß; in which case there will always be (by 226) a fourth vector p, in the same plane, which will represent or construct the function (y: B).a; namely, the fourth proportional to B, y, a. Taking then what we may call the Inverse Index-Functions, or operating on these four vectors a, ß, y, p by the characteristic I-1, we obtain four collinear and right quaternions (209), which may be denoted by v, v', v'', v'"'; and we shall have the equation,

V... v": v= (p: a=y: B =) v": v';

And it is so easy to extend this reasoning to the case of any greater odd number of given vectors in one plane, that we may now consider the recent formula IV. as proved.

289. We shall therefore adopt, as general, the symbolical equations VII. VIII. IX. of 286; and shall thus be enabled, in a shortly subsequent Section, to interpret ternary (and other) products of vectors, as well as powers and other Functions of Vectors, as being generally Quaternions; although they may, in particular cases, degenerate (131) into scalars, or may become right quaternions (132): in which latter event they may, in virtue of the same principle, be represented by, and equated to, their own indices (133), and so be treated as vectors. In symbols, we shall write generally, for any set of vectors a, ß, y,... and any function f, the equation,

=

I... f(a, ẞ, y, ...) = f (I'a, I'ß, I'y, ...) q,

q being some quaternion; while in the particular case when this quaternion is right, or when

q=v = S110 = I1p,

CHAP. I.] PRODUCT OF TWO RECTANGULAR LINES A LINE. 313

we shall write also, and usually by preference (for that case), the formula,

II. . . ƒ (a, ß, y, . . .) = Iƒ (I ̃'a, I-'ß, I ́1y, . . .) = p, p being a vector.

290. For example, instead of saying (as in 281) that the Product of any two Rectangular Vectors is a Right Quaternion, with certain properties of its Index, already pointed out (284, (6.)), we may now say that such a product is equal to that inder. And hence will follow the important consequence, that the Product of any Two Rectangular Lines in Space is equal to (or may be constructed by) a Third Line, rectangular to both; the Rotation round this Product-Line, from the Multiplier-Line to the Multiplicand-Line, being Positive: and the Length of the Product being equal to the Product of the Lengths of the Factors, or representing (with a suitable reference to units) the Area of the Rectangle under them. And generally we may now, for all purposes of calculation and expression, identify a Right Quaternion with its own Index.

SECTION 5.-On some Simplifications of Notation, or of Expression, resulting from this Identification; and on the Conception of an Unit-Line as a Right Versor.

291. An immediate consequence of the symbolical equation 286, IX., is that we may now suppress the Characteristic I, of the Index of a Right Quaternion, in all the formulæ into which it has entered; and so may simplify the Notation. Thus, instead of writing,

The Characteristic Ax., of the Operation of taking the Axis of a Quaternion (132, (6.)), may therefore henceforth be replaced

* Compare the Notes to pages 119, 136, 174, 191, 200.

+ Compare the first Note to page 118, and the second Note to page 200.