1066 
real part of a series of the form: 
aaa, et 1 a, eet ta, ek 4 ....%), deter - (1) 
an J 
where k = k' + k"i with 4" = = and k! =— = (see Comm. N°. 1485, 
§ 4), so that the successive terms represent pure harmonic damped 
vibrations of 3, 5 ete. times the frequency of the main vibration 
and 3, 5 etc. times more rapid damping. *) 
This turned out to be actually the case, at least as far as this 
could be inferred from the observations of the extreme amplitudes *). 
It was found, that within the limits of accuracy of the observa- 
tions the extreme amplitudes could be represented as follows: *) 
1) By a suitable choice of the zero of the time the coefficient a, may be made 
real, but owing to the possible phase-differences the remaining coefficients are in 
that case not necessarily real. 
2) For reasons of symmetry the terms with even powers of k must be absent 
form this series: in fact, a difference between deflections to right and to left 
cannot be made, so that a change of phase of # (increase of ki by (2n +1) =) 
must bring about a change of sign in all the terms. 
3) It may be proved that when the oscillatory motion satisfies equation (1), the 
extreme amplitudes may also be represented by a similar formula, this time with - 
a real value of k (the real part of the complex k) and with real z’s (which, 
however, are not the real parts of the complex z’s). Conversely, if the extreme 
amplitudes can be represented by a formula of the form (1), this will very probably 
also be the case for the complete motion. 
4) This result may be looked upon as a proof, that within the limits of ampli- 
tude of the present experiments the limit was actually reached, below which the 
velocities may be regarded as practically infinitely small. It might perhaps be 
objected that it is only natural that a limited portion of the line 2,¢ may be 
represented by a series of that kind, and that the accuracy which is reached only 
depends on the number of terms introduced; in fact the same would he the case 
with an algebraical series. To this may be answered, that we did not assume 
equation (1) with a definite number of terms a priori and then determined the 
value of the coefficients in the usual way: on the contrary the method of calcu- 
lation actually was such as to show itself the necessity for the use of the formula. 
The method was as follows: a graphical treatment showed that the curve log z, t 
was pretty nearly a straight line which only showed a distinct curvature towards 
the large amplitudes; from the part corresponding to the low a's a first term zj 
could thus be determined with considerable accuracy. The differences «— ”j having 
been drawn up and the line Jog (z—z,),¢ being plotted, it appeared that the line 
obtained, at the lowest limit where it was trustworthy, very distinctly showed 
a direction-coefficient three times as high as the line log -,¢. A first approximate 
value was thus obtained for -3, and this was subtracted from a; in consequence 
of this the line log (-—z,), t became straight over a greater distance than the line 
log a, t and thus a more accurate value for zj could be found and then also 
for zg. If then log (2—z,—zs3),¢t was plotted, the resulting line was found to have 
a five time higher direction coefficient than logs, t‚ etc. 
It should be mentioned that in the calculation the time-intervals between the 
