HAR 
to th: common bafe will be dim‘nifhed in the fame ratio, that 
Nx::CD:C3. For while any arc E G ( fig. 5. 
given in magnitude, let the other 
C FH retains its {pecizs, the line 
nifhed in the fame ratio with C F or CD. 
Cor. 6.—~When the axes C D, C3 (fig. 6.) of two curves 
are very {mall i - Seyi ge eH to their common bafe A B, the 
curvatures at the tops of any two coincident ordinates N K, 
N« are m the ratio of the ordin — — if xp be the 
radius of curvature at x, by have KM x KN 
; whencex N: KN: :KM: —— 
is, as the curvature at x to the curvature a 
C: ence, fuppofing the curve A KDB to have 
the elaiticity and tenfion of 2 mufical chord (fee Cuorn), it 
will vibrate to and fro in curves very nearly of the fame fpe- 
cies with the epek curve AK DB, provided none of the 
dibradiots be too 
For let the firth e e eet of the tenfion son that curve into 
fome other, as A x3 B, in the firft moment of time ; ‘and fince 
are in itor as the curvatures 
at D and K by Cor. 4, and thofe curvatures as the’ ae 
ing forces at D and K, aéting in the direétions DC, K M o 
imi- 
'N very nearly ike thefe forces as the velocities sienatcked 
by them in that fv and the velocities as the oe 
{paces D3, Kx; Pag alternando, DC: D3:: : 
K x,and, dividend D C:3C:: :xN, chuatescthig by 
or. es the curve Ax) B is very nearly of the fame fpecies 
with A DB. And it the next re di it will be chan: 
into Stee of the fame fpecies, and fo on till every point of 
thechord be reduced to the bafe A B at the fame inftant. And 
by the motion here acquired it will bé carried towards the 
oppofite fide of the bafe, till by the oppofition of the ten- 
fion it fhall lofe all its motion by the fame degrees, and in the 
fame curves, by which it was acquired ; and thus the chord 
bntintally. vibrate ih curves of the fame fpecies as the 
firft, negleGing the fmall differeece i in the direCtions KN, 
KM, and the refiftance of the air 
—The final vibrations of a given mufical chord are 
For if the chord at the limit of its vibrations 
afflumes the form of the harmonical curve, it will vibrate to 
and fro in curves of that {pecies by Cor. 7, and its feyeral par- 
ticles. being accelerated by forces conftantly proportional 
to their iiftieces from the bafe A B, will defcribe thofe un- 
equi 
in a cycloid. 
“If the chord at the limit of its vibration. affumes any other 
form, it will cut an harmonical curve, om in length to it, 
in one or more points, as A, K, ig. 7.) 3 and the 
intercepted parts of the chord will be tnore or lets incurvated 
towards 
ater or {maller forces than 
cine oe of the curve : 
and curve to differ in nothing but a3 
erence of the curvatures of the co 
corre 
fuppofing the chor 
ame ow 
» and therefor ore equal to one ariother, : 
Cor. 9. fe 1 op rve 
and its alps of t lane {peciés as the figure of fines. (Sce 
Sixe) Fer fu circle D F QO ( ) ie 5) to grow bi 
ae till it becomes sal toEG R, the figure AKD By wil Se 
-ome @ figure of fines: Becaufe any ordindte K N to the 
iftances in equal times, like’ a pendulum moving 
and therefore to 
therefore, I B 
allo va: 
TLAR 
abfcifs A N, or are GR, bein naenest equat to F 
will then be ‘equal to the fine ot the a ¢ AF ae ce 
every ordinate, as K N, is increafed in ‘be given ratio ja : T 
toC G,orC DtoC E. And, on the contrary, the feve. 
ral ordinates in the “ce figure of fines, diminifhed in 
contftant ratio of C oC D, are the o 
A KDB of it appears, - 
that, as the ordinates are diminifhed in a pager 
portion, their increments and fluxions are diminifhed in 
the fame proportion, the fluxion of the bafe remaining con. 
— ‘ 
—The radius of curvature at the vertex of the “ae 
pe fagitte, which are fimilar portions of the —— 
therefore to each other in the ratio 
shot ae Doig 
1.—The Seer) of which the ordinates are the fums. 
of Be ‘cortefpondin dinates of any two harmonical curves, — 
on equal bafes, but bevlhieg the abfcifs at different points, is 
alfo an harmonical curve. abfcifs of the one curve be- 
ing x, that of the other ee at 2%, st the ordinates 
will be 6. (fin. x) andc. x); now fin.a a= 
(cof. x) . (fin. a) + (cof. sy "thin. ajaudal the jo 
int ordinate 
will be es ec.) (eof. a). (iin. « *) +c. “_ sees a 
if, — be the angle, of which the tangent 
a) Fi its fine and cofine will be in the ratio of 
b+e. = 2) 
c. (fin, a) to 6 4 ¢ (cof. a), and (cof. d) « (fim x) + 
(fin. d) . (cof. x), vl be to the vedi in the conftant ratio 
of fin. dtoc. (fin. a); but (cot d) . (fin. x) + (fin a). 
(cof. a) is the fine of d +x : confequently the newly form 
ed figure is an harmonical curv 4s 
ame may be fhewn ate 2e y 
circles, having shele diameters equal to x greateft ae 
° 
t i 
their centres, this circle will always cut off in the revolving 
line a portion equal to the ordinate. A 
parallel to C D, and EBtoF G, 2A BE= “and 
CHF; but EIB ih as Nell as HE Fy 
ie ge ‘LB: EC7CU Fs: : CH, fince A F is eq ual to 
twice the diftauce of the ae which bi 
Be o' t ET: AE:ID:AG,' 
Chisib; A: C, and the triangles A C CH, DA. 
- lar,and 2 DBI=CHA=DK fs and At 
rallelogram, confequently K D = =e 4 
If the circle CG be fuppofed to pee = 1 a 
interfeQion H will always thew the angular diftanes 
the point in which the curve crofies the axis 5 3 
tance of the centres will be equal to the greateft sitesi ‘ee 
If, therefore, the circles are set the ee ae pina 
are nei. hietes: 
mug: ° mo cal curve. 
the "5 an ag ae 
&. XI. ot s Natural Pao, e “sole oes 
