1905 - 6 .] A New Form of Harmonic Synthetiser. 
225 
the hyperbola and at the other by the line RK or KT as the case 
may be : and take for the value of p the abscissa of the ordinate on 
which the shortest vertical line lies. 
It may easily be ascertained by subtracting (10) and (11) 
respectively from (12) that RKT comes nearer to the hyperbola at 
its apex K than anywhere else, the distance from the one to the 
other being measured vertically. Accordingly the p axis must be 
made to pass through the middle point of KW, and because 
KW has a length equal to r 2 /(2Z), as may readily be proved, while 
the length of UK is Z, therefore 
The abscissa of the point K is r , hence 
p — r ..... (14) 
When t = 0, Q (fig. 6) is at its greatest distance from R, and 
RQ = m - (Z - r). Simultaneously the distance from R of the 
imaginary point given by (tv + r cos t) is iv + r. Hence, calling 
S 0 the distance at the time t = 0 between Q and the imaginary 
point, we have, 
S 0 =m -l - w. 
But from (7) and (13) 
m-w = l + —; 
4 1 
When t = 7r, and Q is at its least distance from R, we have 
similarly, 
The meaning of the above is, that the time axis of the curve 
that would be traced out by our imaginary point on the strip of 
paper passing along at right angles to the line of motion of the 
point, would be parallel to the time axis of the curve traced out 
by the pen on that paper; but the time axis of the point 
would be situated at a distance r 2 /(4l) above the time axis of the 
pen (meaning by the latter a horizontal line drawn along the 
paper half-way between the highest and lowest positions of the pen). 
PROC. ROY. SOC. EDIK, YOL. XXYI. 15 
