1081 
Assuming now that the motion of the body (and of the liquid) is 
a composite harmonic one and damped, corresponding to equation 
(15) and (16) of the previous communication, there are still a great 
number of different conditions of motion possible, which only differ 
by the coefficients o; in this case similarity requires, that those 
coefficients have the same value in all cases, 
3. When the bodies assume such motion not artificially, but by 
friction in the liquids, it will be necessary — in order that the two 
conditions of similarity : equal 7 and equal d may be satisfied — 
that the elements M and A which determine the friction-free motion 
of the bodies also satisfy definite conditions; according to equations 
(17) and (18) of the previous paper these conditions are, that the expres- 
M toe . 
sions = T? and = T have the same values in all cases (k T=-0+4+ 221 
is in all cases the same number). The former condition expresses the 
fact, that the time of swing of the friction-free body Ce =2n: ay 
measured in the time of swing in the liquid as unit, must be the 
same in all cases (conversely —- and this is simpler — the time 
of swing 7’, can in each case be chosen as unit of time). As regards 
7 
Jo tally 
the second condition, according to which each expression ae must 
have the same numerical value in all similar cases, it will be seen, 
having regard to the dimensions of Z,, that this condition requires, 
3 
0 
that the expression 7 has the same numerical value in all cases ; 
in that case all quantities Z, have also equal values in the various 
cases, if each time the specifie units are introduced, because the 
only things left are integrations of functions which possess identical 
‘values in corresponding points, and finally, according to the first 
equations (18), the values o, are also the same in all cases. 
Taken together, the conditions for similarity are therefore, that 
the quantities : 
uk? nRT, 
— and c, = ——— 
nT, K 
or | (2) 
uk nk | 
a and -¢,= 
K ul 
have equal values (in C.G.S. units) in all cases; the same conditions 
as were also found in Comm. N°. 148c. 
