or by what printers call a fuperior letter, fuch as are com-monly used for notes of reference. The pofition and fize of this b, will fufficiently diftinguish it from the note of that name. The afterifm, or if it fhould be thought better, a fmall x, placed before the note, will very well ex-prefs an accidental fharp; and the letter n, a natural; and though these characters for fharps and naturals can never be mistaken for notes, yet for the fake of uniformi-ty, and to prevent all poffible embarraffment in reading, I would place them alfo a little higher than the other letters in the line. Two or more notes founded together, as is frequent on clavicords, harps and other ftringed inftruments, may be expreffed by fetting fuch notes one over the other..

As for the times of notes, which is the fecond thing to be confidered in mufical founds, they may 1 think be very naturally expreffed by the following marks, which are used for a fimilar purpose in common reading.

Semi-demi-quavers, which are fung or played as rapidly as the fyllables of a word are pronounced, may be ex-preffed by the hyphen (which in reading only feparates fyllables) placed after fuch notes. (-)

Semi-quavers, by the comma

Quavers, by the femicolon

Crotchets, by the colon

Miniums, by the period

()

:(i)

(:)

(.)

And femi-breves, by the m dash (—)

A pricked note may be expreffed, as in common nota-tion, by an inverted period fet after the mark fignifying the time of the note.

When two or more notes of equal time come together in the fame bar, the mark of time need only be expreffed after the last of fuch notes, and understood to the reft; and the different fyllables in a bar may, when it is thought: neceffary, be diftinguished by interpofing a space between: fuch fyllables.

Refts

3

Refts will be very well expreffed by the foregoing refpective marks of time fet alone, or not immediately preceded by a note.

A fingle bar may be intelligibly expreffed by the m dash fet fide-wife accross the line. |

A double bar by what printers call a parallel. I

A repeat by a parallel, or the letter S, placed between two colons. :: :S:

All the common marks of graces in mufic may be very well expreffed, or imitated, by the common printing types: thus a trill, by tr, fet above the note; a beat, by the asterisk +, and a turn, by the letter S turned fide-wife, and fet above their refpective notes; a flur, by inclofing the flured notes in a parenthesis ()ftaccato-notes by a period or hyphen fide-wise, set over, or under fuch notes; a pause, by a parenthesis and period together, fet fide-wife, over the note diminutives, or notes of tranfition, as they are not counted in the time of the bar, will be very naturally expreffed by enclosing such notes in a parathefis. []

When three notes are reduced to the time of two, or fix to the time of four, the figure 3 or 6 may be refpectively fet above fuch notes, as in the common notation.

All terms of execution or expreffion as, adagio, andante, piano forte, &c. and figures marking the chords in thorough bafs, may be alfo expreffed as in the common notation.

Explanatory Examples.

Scale of Notes on the G Cliff.

ABCDEFG abc de fgabc d e f g ABCDEFG.

Scale of Notes on the F Cliff.

ABCDEFG abcdefga g a b c d e f g A B C D E F G

Auld Robin Gray. 1ft Mood of common Time, G Cliff, F and C sharp,

:S:

d; | f;'g,a;•b;[b;]a:'a; | b;°g,d;·b‚ba; ;d; | (f;⋅g,)a;'b,[b;]a:g;* f, | e;'d‚g;:f,(f:ed;) { f;'g,a:b;([b;]a:')a; { bb;⋅c,d;'f,[a;]g:e;• e, | (f,a;')d;;f,(g,e;') (c;'d;) | (f;'g,) (e;')f,d: ;'f; | "f:e;'f,d: (f;•g,) | a:ob;•g,(['b;}a:)e;e, | "f;'e,f(*f gf) g*g; | aa'g dg; a: ;a; | f,d;'a;'b,a::a, | bagf;g; ;e; | (f,a;')d,f;'(g,e;')c;'d, | f:e;• d,d: ;f; | gfga babc dAfd afda fdaf edef, d: ;|| 1

Sym.

:S:

tr

No. XVIII.

Obfervations on the Theory of Water Mills, &c. by

Read June 15, 1792.

B

W. WARING.

EING lately requested to make fome calculations relative to mills; particularly Doct. Barker's conftruction, as improved by James Rumfey, I found more difficulty in the attempt than I at first expected. It appeared neceflary to investigate new theorems for the purpose, as there are circumftances peculiar to this conftruction, which are not noticed, I believe, by any author; and the theory of mills, as hitherto published, is very imperfect, which I take to be the reafon it has been of fo little ufe to practical mechanics.

The first step, then, toward calculating the power of any water-mill (or wind-mill) or proportioning their parts and velocities to the greatest advantage, seems to be, The correction of an effential mistake adopted by writers on the Theory of Mills.

This is attempted with all the deference due to eminent authors, whose ingenious labours have justly raised their reputation and advanced the fciences; but when any wrong principles are fucceffively publifhed by a feries of fuch pens, they are the more implicitly received, and more particularly claim a public rectification; which muft be pleafing, even to thefe candid writers themfelves.

George Atwood, M. A. F. R. S. in his masterly trea tife on the rectilinear motion and rotation of bodies, pubifhed fo lately as 1784, continues this overfight, with its pernicious confequences, through his propofitions and corollaries (page 275 to 284,) although he knew the theory was fufpected: for he obferves (page 382) "Mr. Smea

ton in his paper on mechanic power (published in the Philofophical Transactions for the year 1776) allows, "that the theory usually given will not correfpond with "matter of fact, when compared with the motion of ma"chines; and feems to attribute this difagreement, rather "to deficiency in the theory, than to the obftacles which "have prevented the application of it to the complicated "motion of engines, &c. In order to fatisfy himself con

cerning the reafon of this disagreement he conftru&ted a "fet of experiments, which, from the known abilities. "and Ingenuity of the author, certainly deferve great con"fideration and attention from every one who is inter"efted in these inquiries." And notwithstanding the fame "learned author fays, "The evidence upon which the "theory refts is fcarcely lefs than mathematical." I am forry to find, in the present ftate of the sciences, one of his abilities concluding (page 380) "It is not probable that the theory of motion, however inconteftible its principles may be, can afford much affiftance to the practical mechanic," although indeed his theory, compared with the above cited experiments, might fuggeft fuch an inferBut to come to the point, I would juft premise

ence.

these

Definitions.

If a ftream of water imping against a wheel in motion, there are three different velocities to be confidered, appertaining thereto, viz.

First, the abfolute velocity of the water:
Second, the abfolute velocity of the wheel:

Third, the relative velocity of the water to that of the wheel, i. e. the difference of the abfolute velocities ; or the velocity with which the water overtates or ftrikes the wheel.

VOL. III.

T

Now