MATHEMATICAL SECTION. 89 
lita M®rertiIne. FEBRUARY 20, 1884. 
The Chairman presided. 
Eighteen members and guests present. 
Mr. H. Farquaar made a communication on 
EMPIRICAL FORMULZ FOR THE DIMINUTION OF AMPLITUDE OF 
A FREELY—OSCILLATING PENDULUM. 
[Abstract. ] 
The theoretical formule usually employed are obtained by in- 
tegration from an expression for the diminution of the amplitude 
in terms of the amplitude itself. The most important term in this 
expression is one involving the first power of the amplitude, indi- 
cating a resistance proportional to the velocity of the pendulum’s 
motion. A term containing the square of the velocity (or ampli- 
tude) also enters; and, to allow for the friction of the pendulum 
knife-edge on its support, a term independent of the velocity would 
have to be added. Atmospheric resistance to very high velocities 
is found, moreover, to be proportional to a higher power than the 
square of the velocity. There are thus more than three terms the- 
oretically required to express the resistance, and these must be 
calculated, such is the uncertainty of the subject and the complex- 
ity of the conditions on which the different resistances depend, from 
the observations themselves. Since these observations must also 
be depended on for an additional constant (the amplitude at some 
initial time or the time of some standard amplitude), and since 
they are not complete or exact enough to furnish more than three 
constants, or four in a few exceptional cases, it is obvious that a 
good approximation to theory must content us in practice. 
Two convenient methods of representing amplitude in terms of 
time are suggested by imposing arbitrary conditions. First, taking 
three terms to express the diminution (the amplitude being ¢), thus: 
a + be + eg’, 
suppose the square of half the middle co-efficient equal to the 
product of the other two. This expression has then the form: 
= (ep + 6). 
