Intelligence and Miscellaneous Articles. 395 
curve is not seen quite in the same way through the microscope, 
because there the horizontal abscisse are not proportional tu the time 
but to the sine of the time, It must be imagined that the curve 
(fig. 7) is wound up round a cylinder, so that the two ends of 
it meet together, and that the whole is seen in perspective froma 
great distance ; thus it had the real appearance of the curve, as repre- 
sented in two different positions in fig. 8. If the number of vibra- 
tions of the string is accurately equal t to four times the number of the 
_tuning-fork, the curve appears quietly keeping the same position. If 
there is, on the contrary, a little difference of tuning, it looks as if the 
cylinder rotated slowly about its axis, and by the motion of the curve * 
the observer gets as lively an impression of a cylindrical surface, on 
which it seems to be drawn, as if looking at a stereoscopic picture. 
The same impression may be produced by combining, stereoscopically, 
the two diagrams of fig. 8. 
_ We learn, therefore, by these experiments, — 
1. That the strings of a violin, when sleick by the bow, heats 
in one plane. 
_ 2. That every point of the string moves to and fro with two con- 
stant velocities. 
_ These two data are sufficient for finding the complete equation of 
the motion of the whole string. It is the following ; — 
y= Ad 1a ; = )sin(= = ‘) } ah ocainy ste as 
y is the deviation of the point whose distance from one end of the 
string is 7; /, the length of the string; ¢, the time; T’, the duration 
of one vibration ; A, an arbitrary constant; andz, any whole number; 
and all values of the expression under the sign %, got in that way; 
are to be summed. r 
A comprehensive idea of the motion represented by this equation 
may be given in the following way :—Let a 4, fig. 9, be the equili- 
brium position of the string. During the vibration its forms will be 
similar to a ¢ b, compounded of two straight lines, ac and c 6, inter- 
secting in the point c. Let this point of intersection move with a 
constant velocity along two flat circular arcs, lying symmetrically on 
the two sides of the string, and passing through its ends, as repre~ 
sented in fig. 9. A motion the same as the actual motion of the 
whole string is thus given. 
- As for the motion of every single point, it may be deduced froth 
equation (1), that the two parts ad and bc (fig. 7) of the time of 
every vibration are proportional to the two parts of the string which 
are separated by the observed point. The two velocities of course 
are inversely proportional to the times a6 and 6c. In that half of 
the string which is touched by the bow, the smaller velocity has the 
same direction as the bow; in the other half of the string it has the 
contrary direction. By comparing the velocity of the bow with the 
velocity of the point touched by it, I found that this point of the 
string adheres fast to the bow and partakes in its motion during the 
