314 ' H O R O 
little to deftroy that motion. To perpetuate the motion 
of the balance we mu ft therefore have Come exterior force, 
fpme maintaining power, to repair the lpfs of motion ; and 
tiiis power mu ft be in proportion to the greater or lefs re- 
fiftance of the air, and the greater or lets friction of the 
pi vote in their holes. By reducing friction and the refin¬ 
ance of the air to a fmall quantity, the motion of the ba¬ 
lance will be continued by the help of very little repara¬ 
tive power ; and thus the natural vibrations of the balance 
are lefs difturbed, and time more exactly meafured. Our 
objedl therefore is fo to difpofe the balance as to reduce 
friction and the refiftance of the air to the fmalleft quan¬ 
tity poflible; without, however, lofing fight of various 
other properties of the balance neceflary to the correct 
ineafurement of time. 
We are greatly indebted to Monf. F. Berthoud for the 
eftablifhment of found principles on the fubjedl of the ba¬ 
lance, as w*ell as of the pendulum. He it was who firft 
proved that it is not a matter of indifference whether the 
vibrations are quick or flow, the balance large or fmall, 
light or heavy, &c. 
A portable time-keeper, or watch, is expofed to exte¬ 
rior motions or agitations which have an influence upon 
the vibrations of the balance, and upon the extent of then- 
arcs. Suppofing external motion to be in the fame plane 
with the motion of the balance, it is evident that the arcs 
of vibration muft be of greater or lefs extent actcrding 
to the direction of the external motion. The meafure of 
time is moft exadl when the vibrations are equal ; and it 
is neceflary therefore to make them as equal as poflible. 
Perfectly equal, fo as to annihilate the influence of exter¬ 
nal agitations, we cannot make them : but we have the 
means of attaining a very great degree of exadlnefs. 
Suppofe a balance to make two vibrations in a fecond, 
the arc of vibration to be 175 0 , and the external motion 
of the watch, on the fame plane with the watch 25 0 in 
half a fecond: it is evident that the motion of the balance 
is 7 times as ftrong as that of the whole machine, confe- 
quently the arcs of vibration cannot be increafed or dimi- 
niflied more than one feventh: the largelt vibrations then 
will be 200 0 , the final-left 150 0 . But, if we fuppofe the ba¬ 
lance to make four vibrations in a fecond, with arcs of the 
lame extent as before, we fiiall then find that the arcs of 
vibration cannot be increafed or diminifhed more than one 
fourteenth part, for the motion of the balance wili be 14. 
times as ftrong as that of the watch, and the largeft arcs 
•will be 1873, the fmalleft 1635. Upon our firft fuppofition, 
the arcs might vary in extent 50 degrees ; upon our fecond, 
only 25. Hence we may draw this conclufion : “That, 
by increafing the number of vibrations, the extent of the 
arcs remaining the fame, we diminifh the influence of the 
agitations of the watch, or the effedt of exterior motion ; 
and that, generally, the fwifter the balance moves, the 
better it will refift the effects of exterior agitation upon 
the whole machine.” 
Notwithftanding the truth of the foregoing Rule, the 
number of vibrations and the velocity of the balance 
muft be confined within certain limits ; for experience 
has fliown, that extremely-quick vibrations would in- 
creafe fridlion fo much, as prefently to deftroy the differ¬ 
ent parts of the machine. A medium therefore muft be 
obferved ; and experiment has ihown wherein that me¬ 
dium confifts ; namely, “ To allow the balance to make 
not more than 5, nor lefs than 4, vibrations in a fecond.” 
Five vibrations ratty be allowed for machines expofed to 
much agitation; and four for fucli as are merely carried 
in a gentle manner. Our Engl fill chronometers and ma¬ 
rine watches, fo juftly celebrated, commonly make 5 vi¬ 
brations in a fecond, (or 18000 in an hour;) fometimes 
4 only, but never lefs. As to the extent of the arc. of 
vibration, that will vary according to the nature of the 
fcapement. It is always defirable that it fliould reach 
from 180 to 200 degrees: fometimes it will be neceflary 
to have a larger arc; of which we Ihall have occafion to 
ipeak hereafter- 
Before we inquire into the means for diminifliing the 
x 
L O G Y. 
fridlion in the pivots of the balance, it is neceflary to 
fhow what is meant by the quantity of motion in a balance. 
Two things are to be confidered, the weight of the ba¬ 
lance, and its velocity. It is not generally agreed whe¬ 
ther the quantity of motion in a balance be in proportion 
to the product of the weight and the velocity, or in pro¬ 
portion to the weight and the fquare of the velocity; 
M. Berthoud reckons the quantity of motion from the 
fquare of the velocity; and, fince his theory of the ifo- 
chronifm of the fpiral lpring is founded upon this princi¬ 
ple, and fince, by the application of his theory he has 
been able to render equal the vibrations of unequal arcs, 
we think we may venture to aiTume it as a Rule, that 
“ The mafs or weight of the balance multiplied by the 
fquare of the velocity gives the quantity of motion.” 
The quantity of motion of the balance ought to be the 
greateft poflible, regard being had to the friction of its 
pivots; for by the quantity of motion friction is over¬ 
come. Now the lame quantity of motion may be impart¬ 
ed to the balance in two different ways: by a greater- 
weight and lefs velocity, or by a greater velocity and lefs 
weight. Suppofe the weight of a balance to be 16, and 
its velocity 16 alfo : the quantity of motion (by the Rule) 
will be iuX 16 2 — 4096. Suppofing the weight of another 
balance equal to 4 only, and its velocity to 32 : the quan¬ 
tity of motion will be 4X 3^ 2 — 4096, as upon the-firft 
fuppofition. 
We have next to inquire which will molt conduce to 
lefien the friction, increafing the weight of the balance, 
or increafing the velocity of the vibrations. To this end, 
let us fuppofe the velocity of one balance equal to 3, 
and of another equal to 4; the weight of the former 
being 16, and of the latter 9 : it is evident thel'e two 
balances wili haVe the fame quantity of motion; for 
3 2 X 16 = 4 2 X 9 — 144- It is known that the degrees of 
friction are nearly as the fpaces deferibed multiplied by 
the weight of the malfes; confequently the fridlion of 
the firft balance may be expreffed by 3X16* and that of 
the fecond by 4X9- The quantity of motion in the firft 
balance is to the fridlion as 144 to 3X16, or.48 ; and in 
the fecond as 144 to 4X9, or 36. Hence, “There is lei's 
fridlion with a light balance and great velocity than with 
a heavier balance and lefs velocity.” 
Befides this, with a heavy balance it is more difficult 
to regulate a watch, or adjui't it to time, in the different 
pofitions. In the horizontal pofition, the pivots of the 
balance have lefs fridlion, fince the fridlion can take place 
but juft upon the end of the lower pivot which l'upports 
the balance; if the watch therefore be regulated in the 
Vertical pofition, it will gain in the horizontal. In the 
vertical pofition, on the contrary, the fridlion of the pi¬ 
vots of the balance increafes, for both pivots rub through 
their whole length againft the fides of the holes ; and the . 
watch, if it has been regulated in the horizontal pofition,; 
will now lofe time. And it is very evident, that this va¬ 
riation muft be much greater with a heavy balance. Such> 
a balance therefore can be advantageous only in time¬ 
pieces which are never moved out of the horizontal po¬ 
fition. 
It remains to confider whether the diameter of the ba¬ 
lance has any influence upon fridlion. Let us’then fup¬ 
pofe two balances of equal weight, but of different dia¬ 
meters, their extremities deferibing l'paces of equal ex¬ 
tent. . (By equal fpaces we do not mean the fame num¬ 
ber of degrees of a circle; for v if the diameter of one ba¬ 
lance were double that of the other, to deferibe equal 
fpaces'the fmaller balance muft pafs through twice as 
many degrees as the large one.) Now it is evident, that,, 
though the extent of fpace be the fame for each balance, 
yet the pivots of the large balance will move through lefs. 
fpace. than thofe of the fmall one; that is to fay, only 
half the fpace if the larger balance be double the diame¬ 
ter of. thd final] one. It is proved alfo, that, “ the fric¬ 
tions are in proportion to the fpaces deferibed and the - 
weight of the malfes.” The weight being, upon our fup¬ 
pofition, the fame, the faddons are as the fp.aces de¬ 
feribed 5 
