254 
DR. J. YOUNG- AND PROFESSOR G. FORBES ON THE 
graph. In the short interval of time (usually about one-fourth or one-fifth of a second) 
between two wheel marks this retardation does not affect us. We can say, in the 
above example, that the cylinder is passing over 0*2411 revolution in 100 turns of 
the toothed wheel, at the instant when the pen is at the reading l,-437 ; and that it is 
passing over 0*2404 revolution in 100 turns of the toothed wheel when the pen is at 
the reading 5,046. A simple proportion tells us that when the pen is at the reading 
of the signal, i.e ., 3,810, the number of revolutions in 100 turns of the toothed wheel is 
0*2411 —HtjJX 0*0007=0*2406. 
But in the case of the clock trace we use intervals of two seconds. Here the case is 
very different. From the readings of the chronograph we must determine what was 
the actual velocity of the cylinder at some particular reading of the chronograph, what 
was the retardation produced by the excess of friction, what is the law according to 
which the velocity of rotation of the cylinder varies with the reading upon the 
cylinder; and thence what was the velocity of its rotation at the time when the signal 
was indicated. 
Let s be the reading at any time t, arid v the velocity of rotation of the cylinder. 
When £=0, let s=s 0 , and let v—u. 
Let s=s l and s 2 when the times are t 2 respectively. 
Let f be the excess of friction, or the retardation. 
Then 
«i=So+“ J i— 1/V 
% h— u (h *i) ^i 2 ) 
v ^r u ~ f 2 
Therefore v=the velocity of rotation at the time 
Now let s be the reading corresponding to the velocity of rotation v, or the reading 
at that time ttfi. 
Therefore 
