tesimally imaginary. Though this course may seem to savour of the system "ignotum per ignotius," it is no less true than singular that in this instance a difficulty more peculiarly affecting reals receives light from the consideration of imaginaries. To exhibit therefore in its most general form admitted in algebra, including imaginaries as well as reals, let

x = y + √=1%

(5.) y and %, which I call the "constituents" of x, being independent quantities, positive, negative, or + 0 or -0. Then we shall have

Here it is important to remark that the second constituent of x is always of a different sign from the corresponding con

I must now lay down some necessary definitions of the notation employed, preparatory to the statement of a proposition which will be found further on, and I must here add that, in resorting to a new specificatory or individualizing notation, I have unwillingly yielded to necessity, from finding that the indeterminateness of ordinary exponential and inverse trigonometrical expressions almost always occasioned perplexity and frequently led into material errors of reasoning; not that I was unaware of the repugnance which any new notation has to encounter, and the increased difficulties it opposes to the reception of any new theory. Let the symbol, placed before a positive quantity, be appropriated to denote the positive square root; then will denote 1 or

x

-1, according as ≈ is positive or negative. When a is real, and therefore = 0, we have no more right to consider z positive than negative, unless it be known absolutely that the variable x is in such a varying state as to be about to become x+dy + -1d z, the sign of the real infinitesimal d z being known; but still, when a is real, and nothing else is known with reference to its varying state, this at least we know alternatively, that if we consider = 1, there would be a √ 22

2

When once we introduce into our calculations the consideration of imaginary values, we have to treat a ory + -1%, as essentially a variable in both its constituents, and therefore ≈ can never fairly be supposed in the condition of an absolute central stationary zero. Let y2+2 denote the arithme

-1

y

tical Neperian logarithm of √y+. Let coso + y + z2 denote the circular arc not less than +0 and not greater than , which, when radius=1, is equal to It is ob

y

vious that such an arc is always assignable, since y

can never be greater than 1, nor less than -1.

I come now to the following proposition, before referred to, viz.

y

is a Neperian

" I √ y3 + x2 + √ COS 1 v logarithm or e-log of x." This proposition is proved in my papers on exponential functions cited in the 8th volume of this Magazine, p. 281, (Number for April 1836); but as the heads of the proof are very simple and at the same time illustrate my subsequent reasoning in the present case, I shall briefly recapitulate them here.

Let the notation e denote that particular value of the ambiguous expression e, which is represented by the sum of the

where is not limited to real values. It has not been unusual to treat of e' when is imaginary, and when this has been done, the meaning of e must have been tacitly defined (though probably without regard to the possibility of eʻ having many values for any given ) by reference to the preceding series, which is convergent for all values of 8. A series, when resorted to as a definition is most convenient if always converging, but in development a series is not to be considered as correct and safe merely on account of its convergence, for expressions may be assigned which are developable by some incomplete methods in a converging series, and yet may be shown, from the functional properties which constitute their best definition, to be equal to n terms of such series together with a remainder (sometimes representable in the form of a definite integral,) which, in certain cases, instead of approaching towards 0 as a

limit like the remainder of the series, recedes from 0 as n increases. It is demonstrable from the nature of the series (7.), and may, I think, be assumed as an admitted proposition, that

Now, of the right-hand member of (9.), the first factor, viz. ly2+ is evidently equal to y2+x2, and if it can be

shown that the other factor, viz. eo

.I
COS

0+ 3+ is

our proposition will be proved, for

COS

it will be proved that e++ cos

is an e-log of y +

1

0+

-1x or x, according to the most limited

definition I can conceive of the term Neperian logarithm that

will extend to imaginaries.

if by cos be understood the sum of the series

(10.)

(11.)

(12.)

This not only follows immediately from the definition above given of the notation eo, but the definitions of cos and sin 9 accord with admitted theorems respecting the sine and cosine when is real. The two series (11.) and (12.) are convergent for all finite values of 0, and I can see no objection to them as definitions of sine and cosine, even when is imaginary. I do accordingly treat them as such in my general exponential theory,

but for our present purpose it is enough that equation (10.) be admitted when is real.

It follows from that equation that

y

(14.)

0+ √ y2 + x2 √ y2 + 22

for an arc, whether positive or negative, has the same cosine,

for if an arc be respectively in the first positive or in the first negative semicircle, the sign of the sine of that arc will be positive or negative accordingly.

At this step it is important to remark the necessity that ex

ists for the introduction of the expression, in order to

N=1% 3+ √ = 1 z Y+

Ny3 +22

in all cases, for if that expression were omitted,

and therefore sin (cos+T)

COS

Ny2

y

substituted for

we should not have

sin (cos +
2o+ √ y2 + x2

unless z happened to be positive; for since the sum of the squares of the sine and cosine of any arc is equal to 1, and since the sign of the sine of an arc in the first semicircle is al

LXVII. Experimental Researches into the Physiology of the Human Voice. By JOHN BISHOP, Esq., &c. &c.

[Continued from p. 277, and concluded.]

THE IE falsetto, or voce di testa, has always been considered a most embarrassing subject of research, and its peculiar quality has excited the attention both of the physiologist and of the musician. The change produced in the voice when passing from the falsetto into the common tone, or the reverse, is in some persons very sensible to the ear, whilst in others it is almost imperceptible. It is remarkable that some individuals have the faculty of producing, in the same pitch, three or four tones, possessing either the falsetto or the common character, a circumstance which indicates that the difference between them depends rather upon an altered state of the vocal tube than upon any change in the glottis.

The falsetto has generally been ascribed to some particular adaptation of the upper ligaments of the larynx. Dodart * has attempted to prove that it is a supra-laryngeal function, and that the nose becomes the principal tube of sound instead of the cavity of the mouth. Bennati† also considers these tones to be modified by the supra-laryngeal cavity, an opinion not justified by the experiments which he has detailed.

According to this hypothesis, we must suppose the influence of the trachea to be entirely annulled; but on what acoustic principle this is to be effected he does not explain, nor indeed can any one else. The changes observed by him in the pharynx were undoubtedly associated with corresponding changes in the whole length of the tube, and all the phænomena he has described may thus be readily explained.

It was suggested to me by Mr. Wheatstone, that it was only necessary to suppose the vocal tube capable of subdividing its vibrating length to account for this peculiar character of tone. Analogous effects are observed in the clarionet, the flute, and other instruments; the change taking place at the twelfth of the fundamental note in the former instrument, and at the octave in the latter. Having had an opportunity of examining the phænomena in some individuals possessing remarkably fine voices, I placed my finger lightly on the larynx, and requested them to gradually elevate the voice from the primary to the falsetto tones, when, although the ear could scarcely distinguish

* Mém. de l'Acad., 1707.

† Recherches sur le Mechan. de la Voix Hum.