1078 
In this manner it becomes intelligible that the oscillating body, also 
when it is made to swing freely in the liquid, will assume a motion 
corresponding to equation (16)'), as was first revealed by experiment 
(comp. Part. VI). 
In equation (16) the exponent & which contains the time of oscil- 
lation 7’ and the logarithmic decrement (e=) as also 
the coefficients 6,,6,, etc. (a is arbitrary) are determined by the 
interactions between liquid and body. By the friction to which the 
body is subject, moments act on it which can be calculated, as was 
actually done in the first approximation in Comm. N°. 1485 for the 
case of a sphere, as soon as the functions wn and y„ are known; 
if the moments of the couples caused by the successive sets of motion 
are represented by C,, C,, C, etc. (the moments of even order are 
all zero), the equation of motion of the body is: 
da 
dt? 
(comp. 23, Comm. N°. 1485), where C= CC, + C, + C, +... The 
quantities C are given by 
EEM Oel ven er lan 
~ Zz = \ 
2 2 
dw, op da 
C= —f orn = „fe Se ds °) = ynkay | O° a ds = — IL, oe : 
zi zi zi 
where (18) 
Fn 
ie uf 0° IN dr AS 
A, is a numerical quantity (of the dimensions of a volume) which 
depends on the shape of the body and further on the quantity 5, 
i.e. on £ (time of swing and decrement), on the constants n and u 
of the liquid and finally on the coefficients o up to and including 
6; °). Equation (17) can thus be separated into a series of equations 
t) That is to say after the disturbances, which are due to the starting of the 
motion, have subsided: these disturbances are not gone into here (comp. Comm. 
N°. 1485, § 4, note). 
*) As in Comm. NO. 1485, F represents the tangential force per unit area in the 
direction of the motion; ds is the area of a circular strip round the body of 
a is the gradient of the angular velocity of the nth order in the 
liquid close to the body; 2, and 23 are the z-limits of the body. 
5) If different parts of the oscillating system are surrounded by different fluids 
(e.g. a part by a liquid and the other part by air, as was the case in the 
experiments) Ln itself has to be divided into parts, each of which refers to one 
of the fluids. 
revolution ; 
