rust E. 
always as the e produ& or re€tangle of the lines C F x 
this given ce ity may be made = a 4, in which cafe we “hall 
have FG = a To determine, therefore, the equation 
of the curve B D, let KH = =) HG= 
and GC = y; then heel of the fimilar triangles H Ky I 
and GK F, wehave HK: H1I::GK:FG = “Ss that 
ab 
is, at: a::a + «:—3; whence we have aazay+nry, 
which is the equation of the curve, and which fhews it to 
be that of the 4 dae with a to fi ‘sa ages 
the curve and its afymptote en 
a=y, or HK = HD; alfo when es point ce ar ae - 
A, theny = AB; andbecaulleEA x AB=IHxH 
wehave EA:1H::HD: AB: eer the great- 
ef force of the {pring be the fulee to its — force. Again, 
becaufe the ordinate 
e true om "Of the folid. or fuafee is is 
generated, as feen Pe Te 
ur jigs 7. one the fubftance of three figures, as 
given by Maran, and the letters of reference are altered, 
fo as to prevent the recurrence o ame letters, eae, 
in the as nal account rendered the a ee = 
guous to the 
many oppofite eed eee oe aferibed Sue: the 
afymptotes » £, and o x interfecting one another in the centre ¢. 
Put eo F=o8] = 1, theneB = V 2 = radius of the cir- 
cle 3 3 é A touching the al te igre in ee re- 
{pective vertices: again, v2 = 2 is 
the diftance of the focus of each hyperbola from de centre ; 
and laftly, the diagnos a 
agreeably to 
fore plain, that the fectio 
its axis 
two equal and adjacent hyperbolas, beginning from their 
vertices 6 and 9 It is alfo evident, that 
made, of the 
ie aft ends ena er be exa actly proportioned to th the leait 
and greatelt force of the {pring reciprocally. 
Tt alfo follows, that when the pe per on of the greateft 
and. leaft force of the {pring is known, or the ratio of o @ 
aa of the fufee » « isa given 
a 
ae ngth of the fu 
ry are given, then the other correfponding 
at ca and not to be affumed at aE afure. 
Having thus determined the geometrical form of the fufee, 
it may not be unacceptable to the reader to give an illuftra- 
tion of this peel by a of — been meee ri 
us therefore put = hy VG yy n 
the equation aa=ay fe ae - will on in ite"t proper ee 
cae 
“for ce is 
whence if a and y be given to find x, we have —— —a= 4; 
when x is given, or x = 1, we have given the eee . ato ys ; 
fer then aa =ay+ y3 confequently ary: 
= y; and a. when 
a 
When @ and » are given, then ma 
a 
a and y are given, we have « a —ay= wy, and, completing 
the fquare, a= xy + Fy? + 4. 
The proper denominations for exprefling the refpective 
forces, a and ys of the main-fpring, may be ounces and frac- 
5. 
many tenth parts of an an in the meafures ¢ Ff, o yO 
The a forces ed be cel uelane ibe let 
o 
ee Ades he barr aeons oan a 
r gu 
ail ey U, fo as to ae. over it, ee to fu ort the weight 
V, which is juft large enough to balance the {malleft force 
of the panes thencB=a:va=xzy 
Ca 1 Suppole the weight S . be 63 Bunees, and the 
weight V to be then a = 63 a =2 31 5 -ar, —— 
is the fame fans, we = 3 andy = 13 fae we find a = 
a 5 
— —a= 6= 24, that is, when the extreme forces are 
3: 1, the length of the fufee yo is 7 te oe canes of 
V: oR: 
its bafe or hee end 86. When S: ty 
7 hen 7 = a, B; alfo when S a 3 £2, then 
=a; a univertlly if ae :V: Daisy: 
a+: a, then it will be = ones 
—Given the length of the fue o cv 6,t ter- 
Ca 
te ratio of the forces or ee > VW, which he Ae 
the diameters @ @ and 1 ae ends of the fufee 
Since a =ay He 6 E pens to the laws of the hyperbo- 
la, we have a: + 6:4; then Pe affuming the value 
of awe i that ofy ne fippofe a = 3, then aryit3 
+ and in this Kier the dence 6 8 is three 
times ne ee nas and e m:n, therefore 
thus if m: 2 
for any affumed ratio, we have - tema; 
n 
h =aso ee :2, then 2” = a3 
ue if « = 6, we have a = Ta, and y = 8. 
afe 3.—Given the length of the fufee, and the greatelt 
force of the main-fpring to find the leaft force of the os 
Suppofe#=o¢yv= 6, anda =o = 3, then y = <5 
a 
stag 3; fo that if S = 63 ounces, we have V = 213; and 
if « = a, then rOg 
Cafe ae ns ae = [ fates, and the leaft force 
of he main-{pring, to find the greatett. . = 6, y 
= 1, thn Vey tiv +hy = 7 6.25 + oF = 3 
= a; fo via if y = 21 ounces, es or ‘anh force will be 
= 64: Baelore. 4 in every cafe, the form and dimenfions 
of the alee 1 are, or may be, ee ei determined. 
But we have yet to fhew how the requifite hyperbola « 6 y. 
may be defcribed, as a pattern for the curve of the fufee.. 
Suppofe the extreme forces 
Qe Tek, continued each ray indehintely Then having de- 
termined the diamete 7 he bafe 6 8 of the he take 
that extent in the divider fom fet it way: from the 
centre ¢ in the line 7 « i and ?, which points will be the 
oci of the two oppofite hy has aByand de Tn 
the focus ¢ fix a ruler g x J, fo as to be moveable round a 
pin as a centre at ree ney heey as much fhorter- 
