i66 Mr. Atwood's Investigations for determining 
— O Q (fig. 8 and 5.) from the point of quiescence cf the ba- 
lance spring, as corresponds with the error intended to be rec- 
tified .* In like manner, if the property of the balance spring 
should be such that a diminution of the arc of vibration causes 
an acceleration of the daily rate, this error will be corrected by 
placing the points of quiescence of the auxiliary springs in the 
first semiarcs of vibration, (fig. 4 and 8.) at their proper dis- 
tances from the point of quiescence of the balance spring. It 
is, however, to be remarked, that if the balance spring should 
be of the latter kind here' assumed, and the points of quies- 
cence of the auxiliary springs, either by any casual derange- 
ment in their position, or by an adjustment of them for the 
purpose of experimental observation, should be placed in the 
latter semiarcs of vibration, the effect of this position would 
be an acceleration of the rate, whenever the semiarc of vibra- 
tion is diminished : and this effect would be produced on a 
double account ; first, from the assumed nature of the balance 
spring, which disposes it to describe the smaller arcs in less time 
all the circumstances capable of influencing the times of vibration in longer or shorter 
arcs, were either not noticed, or omitted to be properly allowed for ; this will seem the 
more probable if it be flowed, that a balance spring may be adjusted in various ways 
so that either the longer or shorter arcs shall be performed in the least time ; not only 
by altering the thickness and strength of the spring in different parts, but, if we sub- 
scribe to the opinion of M.Berthoud, in a spring uniform in every respect through- 
out, by altering the length and number of turns. He infers, that a certain length and 
number of turns may be given to an uniform spiral spring which will make it perfectly 
isochronal. This latter principle, however, does not appear to have been verified by any 
satisfactory experiments. According to the inferences deduced from the preceding in- 
vestigations, three spiral springs, which are not isochronal when acting singly, may be 
so united by properly adjusting their points of quiescence, that their combined action 
shall cause the balance to perform its vibrations in any two arcs of unequal lengths m 
the same time. . , 
* The position of the points of quiescence of the auxiliary springs is here un erstoo 
to be altered, by affixing the arms G O, I K, differently on the axes T R> F S ; the 
quiescent position of the pallets being no ways changed. 
