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CHAP. 1.]



(11.) In like manner, by 192, II., it may be inferred that

XV... K'qq'q = K(q′′. q′q) = Kqʻq. Kq" = Kq. Kg. Kq",

with a corresponding result for any greater number of factors; whence by 192, I., if IIq and II'q denote the products of any one set of quaternions taken in two opposite orders, we may write,

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(12.) But if v be right, as above, then Kv=- v, by 144; hence,

XVIII. . . KIIv=±ÏI'v; XIX... São = + SI ́v; XX... VIIv=VII'v; upper or lower signs being taken, according as the number of the right factors is even or odd; and under the same conditions,

XXI... SIIv=}(IIv + II ́v) ;,

XXII... VIIv = { (II v‡ I'v) ;

as was lately exemplified (1.), for the case where the number is two.
(13.) For the case where that number is three, the four last formulæ give,
XXIII... So'v'v = − Svv'v′′ = 4 (v′′v'v — vv'v′′) ;

XXIV. . . V v′′v'v = + Vvv'v′′ = } (v ̈vv + vv'v′′);

results which obviously agree with X. and IV.

224. For the case of Complanar Quaternions (119), the power of reducing each (120) to the form of a fraction (101) which shall have, at pleasure, for its denominator or for its numerator, any arbitrary line in the given plane, furnishes some peculiar facilities for proving the commutative and associative properties of Addition (207), and the distributive and associative properties of Multiplication (212, 223); while, for this case of multiplication of quaternions, we have already seen (191, (1.)) that the commutative property also holds good, as it does in algebraic multiplication. It may therefore be not irrelevant nor useless to insert here a short Second Chapter on the subject of such complanars: in treating briefly of which, while assuming as proved the existence of all the foregoing properties, we shall have an opportunity to say something of Powers and Roots and Logarithms; and of the connexion of Quaternions with Plane Trigonometry, and with Algebraical Equations. After which, in the Third and last Chapter of this Second Book, we propose to resume, for a short time, the consideration of Diplanar Quaternions; and especially to show how the Associative Principle of Multiplication can be established, for them, without* employing the Distributive Principle.

* Compare the Note to page 236.




SECTION 1.-On Complanar Proportion of Vectors; Fourth Proportional to Three, Third Proportional to Two, Mean Proportional, Square Root; General Reduction of a Quaternion in a given Plane, to a Standard Binomial Form.

225. The Quaternions of the present Chapter shall all be supposed to be complanar (119); their common plane being assumed to coincide with that of the given right versor i (181). And all the lines, or vectors, such as a, ß, y, &c., or a0, a1, a2, &c., to be here employed, shall be conceived to be in that given plane of i; so that we may write (by 123), for the purposes of this Chapter, the formulæ of complanarity:

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9 ||| ' ||| ... ||| ¿; a ||| i, ao || i, &c. 226. Under these conditions, we can always (by 103, 117) interpret any symbol of the form (B: a).y, as denoting a line & in the given plane; which line may also be denoted (125) by the symbol (y: a).ẞ, but not* (comp. 103) by either of the two apparently equivalent symbols, (ẞ.y): a, (y.ẞ): a; so that we may write,

I... 8 = B y = 1 ß,
= 1B,


and may say that this line 8 is the Fourth Proportional to the

* In fact the symbols B. y, y. ß, or ẞy, yẞ, have not as yet received with us any interpretation; and even when they shall come to be interpreted as representing certain quaternions, it will be found (comp. 168) that the two combinations, Y and



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have generally different significations.




three lines a, ẞ, y; or to the three lines a, y, B; the two Means, ẞ and y, of any such Complanar Proportion of Four Vectors, admitting thus of being interchanged, as in algebra. Under the same conditions we may write also (by 125),

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so that (still as in algebra) the two Extremes, a and 8, of any such proportion of four lines a, ß, y, d, may likewise change places among themselves: while we may also make the means become the extremes, if we at the same time change the extremes to means. More generally, if a, ß, y, d, e... be any odd number of vectors in the given plane, we can always find another vector p in that plane, which shall satisfy the equation,

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and when such a formula holds good, for any one arrangement of the numerator-lines a, y, ε, and of the denominator-lines p, B, 8... it can easily be proved to hold good also for any other arrangement of the numerators, and any other arrangement of the denominators. For example, whatever four (complanar) vectors may be denoted by Byde, we have the transformations,

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so that the two denominators also may change places.

227. An interesting case of such proportion (226) is that in which the means coincide; so that only three distinct lines, such as a, ẞ, y, are involved: and that we have (comp. Art. 149, and Fig. 42) an equation of the form,

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but not* y = BB: a, nor a = ßß: y. In this case, it is said that the three lines aẞy form a Continued Proportion; of which a and y are now the Extremes, and B is the Mean: this line ẞ being also said to be at Mean Proportional between the two others, a and y; while y is the Third Proportional to the two lines a and ẞ; and à is, at the same time, the third proportional toy and B. Under the same conditions, we have

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so that this mean, B, between a and y, is also the fourth proportional (226) to itself, as first, and to those two other lines. We have also (comp. again 149),

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although we are not here to write ẞ= (ya), nor ẞ= (ay). But because we have always, as in algebra (comp. 199, (3.) ), the equation or identity, (− q)2= q2, we are equally well entitled to write,

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the symbol q* denoting thus, in general, either of two opposite quaternions, whereof however one, namely that one of which the angle is acute, has been already selected in 199, (1.), as that which shall be called by us the Square Root of the quaternion

Compare the Note to the foregoing Article.

We say, a mean proportional; because we shall shortly see that the opposite line, - ß, is in the same sense another mean; although a rule will presently be given, for distinguishing between them, and for selecting one, as that which may be called, by eminence, the mean proportional.


q, and denoted by q.

We may therefore establish the for



VII. B = ± √(2). α =

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if a, B, y form, as above, a continued proportion; the upper signs being taken when (as in Fig. 42) the angle aoc, between the extreme lines a, y, is bisected by the line oв, or ß, itself; but the lower signs, when that angle is bisected by the opposite line, -ẞ, or when ẞ bisects the vertically opposite angle (comp. again 199, (3.)): but the proportion of tensors,


VIII... Ty: TB TB: Ta,

and the resulting formulæ,

IX... Tẞ2 = Ta. Ty,

TB = √(Ta.Ty),


in each case holding good. And when we shall speak simply of the Mean Proportional between two vectors, a and which make any acute, or right, or obtuse angle with each other, we shall always henceforth understand the former of these two bisectors; namely, the bisector Oв of that angle AOC itself, and not that of the opposite angle: thus taking upper signs, in the recent formula VII.


(1.) At the limit when the angle AOC vanishes, so that Uy Ua, then Uẞ= each of these two unit-lines; and the mean proportional ẞ has the same common direction as each of the two given extremes. This comes to our agreeing to write, X'... √(a2) =+ a,

X... V1+ 1, and generally,

if a be any positive scalar.

(2.) At the other limit, when AOC=, or Uy =- Ua, the length of the mean proportional ẞ is still determined by IX., as the geometric mean (in the usual sense) between the lengths of the two given extremes (comp. the two Figures 41); but, even with the supposed restriction (225) on the plane in which all the lines are situated, an ambiguity arises in this case, from the doubt which of the two opposite perpendiculars at o, to the line AOC, is to be taken as the direction of the mean vector. To remove this ambiguity, we shall suppose that the rotation round the axis of i (to which axis all the lines considered in this Chapter are, by 225, perpendicular), from the first line oa to the second line OB, is in this case positive; which supposition is equivalent to writing, for present purposes,

XI. * . . √ − 1 = +i; and XI'... V(−a2)=ia, if a >0.

It is to be carefully observed that this square root of negative unity is not, in any sense, imaginary, nor even ambiguous, in its geometrical interpretation, but denotes a real and given right versor (181).

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