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(1.) In connexion with the convergence of this ponential series, or with the inequality III., it may be remarked that if we write (comp. 235) r = Tq, and rm = Tqm, we shall have, by 212, (2.),
XI. . . T(P(q, m + n) − P(q, m) ) ≤ P(r, m+ n) − P(r, m);
it is sufficient then to prove that this last difference, or the sum of the n positive terms, rm+1, Tmin, can be made <a. Now if we take a number p>2r - 1, we shall have rp+1 <&rp, Tp+2< }rp+1, &c., so that a finite number m>p>2r - 1 can be assigned, such that <a; and then,
XII. . . P (r, m + n) − P(r, m) < a(2-1+2-o + . .' + 2-") < a;
the asserted inequality is therefore proved to exist.
(2.) In general, if an ascending series with positive coefficients, such as XIII... Ao + A19 + A292 + &c., where Ao>o, A1 > 0, &c.,
be convergent when is changed to a positive scalar, it will à fortiori converge, when q is a quaternion.
240. Let q and q' be any two complanar quaternions, and let q' be their sum, so that
then, as in algebra, with the signification 239, II. of qm, and with corresponding significations of q'm and q'm, we have
where q%=q%=1. Hence, writing again r=Tq, rm=Tq, and in like manner r' = Tq', r'' = Tq'', &c., the two differences,
III... P(r', m).P(r, m) - P(r", m),
IV... P(r", 2m) – P (r', m).P(r, m),
can be expanded as sums of positive terms of the form r'.", (one sum containing m (m + 1), and the other containing m(m + 1) such terms); but, by 239, III., the sum of these two positive differences can be made less than any given small positive scalar a, since
V... P(', 2m) - P(r", m) <a, if a>0,
provided that the number m is taken large enough; each difference, therefore, separately tends to 0, as m tends to ∞; a tendency which must exist à fortiori, when the tensors, r, r', r'', are replaced by the quaternions, q, q', q". The function Pq is therefore subject to the Exponential Law,
VI... P(q'+9)= Pq'. Pq = Pq. Pq', if q'q.
(1.) If we write (comp. 237, (5.) ),
VII... P1 = ε, then VIII... Px = (e)。 = arithmetical value of ε;
where is the known base of the natural system of logarithms, and x is any scalar. We shall henceforth write simply & to denote this principal (or arithmetical) value of the xth power of ɛ, and so shall have the simplified equation,
VIII'... P= ε.
(2.) Already we have thus a motive for writing, generally, IX... Pq = ε;
but this formula is here to be considered merely as a definition of the sense in which we interpret this exponential symbol, ɛ; namely as what we have lately called the ponential function, Pq, considered as the sum of the infinite but converging series, 239, X. It will however be soon seen to be included in a more general definition (comp. 238) of the symbol q'.
(3.) For any scalar x, we have by VIII. the transformation:
X... x=1Px=natural logarithm of ponential of x.
241. The exponential law (240) gives the following general decomposition of a ponential into factors,
I... Pq= P(x+iy) = Px. Piy;
in which we have just seen that the factor Px is a positive scalar. The other factor, Piy, is easily proved to be a versor, and therefore to be the versor of Pq, while Px is the tensor of the same ponen- . tial; because we have in general,
II... Pq.P(-q)= P0=1, and III... PKq=KPq, IV... (Kg) K (9") = (say) Kq" (comp. 199, IX.); and therefore, in particular (comp. 150, 158),
VI... NP1y = 1. Hence?.
VIII... x = Sq=1TPq;
IX... UPq=PVq=Piyev cis y (comp. 235, IV.);
this last transformation being obtained from the two series,
Hence the ponential Pq may be thus transformed:
XII... Pq = P(x + y) = e' cis y.
CONNEXION WITH TRIGONOMETRY.
(1) If we had not chosen to assume as known the series for cosine and sine, nor to select (at first) any one unit of angle, such as that known one on which their validity depends, we might then have proceeded as follows.
(− y) = - oy,
XIV... f(y+y)= S(Piy. Piy')=fy.fy' - oy.oy';
and then the functional equation, which results, namely,
would show that
XVI. . . ƒ (y + y') + f (y −y') = 2fy.fy',
XVII... fy = cos(2× a right angle }
whatever unit of angle may be adopted, provided that we determine the constant c by the condition,
XVIII... c = least positive root of the equation fy(= SPiy) = 0 ;
XVIII'... c = 1.5708, as the study of the series* would show.
(2.) A motive would thus arise for representing a right angle by this numerical constant, c; or for so selecting the angular unit, as to have the equation ( still denoting two right angles),
XIX. . . π = 2c = least positive root of the equation fy = -1;
XXI... (fy)2+(py) Piy. P(-iy) = 1,
it is evident that øy =+ sin y; and it is easy to prove that the upper sign is to be taken. In fact, it can be shown (without supposing any previous knowledge of coSines or sines) that pc is positive, and therefore that
* In fact, the value of the constant c may be obtained to this degree of accuracy, by simple interpolation between the two approximate values of the function ƒ,
and of course there are artifices, not necessary to be mentioned here, by which a far more accurate value can be found.
(4.) The series X. XI. for cosine and sine might thus be deduced, instead of being assumed as known and since we have the limiting value,
XXIX... lim. y sin y = lim. y ̄1i ́1VPiy= 1,
it follows that the unit of angle, which thus gives Piy = cis y, is (as usual) the angle subtended at the centre by the arc equal to radius; or that the number π (or 2c) is to 1, as the circumference is to the diameter of a circle.
(5.) If any other angular unit had been, for any reason, chosen, then a right angle would of course be represented by a different number, and not by 1·5708 nearly; but we should still have the transformation,
242. The usual unit being retained, we see, by 241, XII., that
I... P. 2in1, and II... P(q+2in) = Pq,
if n be any whole number; it follows, then, that the inverse ponential function, P1q, or what we may call the Imponential, of a given quaternion q, has indefinitely many values, which may all be represented by the formula,
while the one which corresponds to n=0 may be called the Principal Imponential. It will be found that when the exponent p is any scalar, the definition already given (237, IV., XII.) for the nth value of the pth power of q enables us to establish the formula,
V... (q) = P(pP ̈1q);
and we now propose to extend this last formula, by a new definition, to the more general case (238), when the exponent is a quaternion q': thus writing generally, for any two complanar quaternions, q and q', the General Exponential Formula,
VI... (9) = P(q'P, 1q);
the principal value of q" being still conceived to correspond to n = 0, or to the principal amplitude of q (comp. 235, (3.)).
LOGARITHM OF A QUATERNION.
(1.) For example,
VII. . . (ε9)o = P(qPo ̄1ɛ) = Pq, because Po1 = le = 1;
the ponential Pq, which we agreed, in 240, (2.), to denote simply by ea, is therefore now seen to be in fact, by our general definition, the principal value of that power, or exponential.
these two last only differing from the usual imaginary expressions for cosine and sine, by the geometrical reality* of the versor i.
(3.) The cosine and sine of a quaternion (in the given plane) may now be defined by the equations:
(4.) With this interpretation of cis q, the exponential properties, 236, IX., X., continue to hold good; and we may write,
XII. . . (qa′)n = P (qʻlTq). P (iqʻ am, q) = (Tq)02′ cis (q ́am, q);
a formula which evidently includes the corresponding one, 237, IV., for the nth value of the pth power of q, when p is scalar.
(5.) The definitions III. and VI., combined with 235, XII., give generally,
which last equation agrees with a known interpretation of the symbol,
considered as denoting in algebra a real quantity.
(7.) The formula VI. may even be extended to the case where the exponent q'is a quaternion, which is not in the given plane of i, and therefore not complanar with the base q; thus we may write,
but it would be foreign (225) to the plan of this Chapter to enter into any further details, on the subject of the interpretation of the exponential symbol q', for this case of diplanar quaternions, though we see that there would be no difficulty in treating it, after what has been shown respecting complanars.
* Compare 232, (2.), and the Notes to pages 243, 248.