1905-6.] A New Form of Harmonic Syntheliser. 
223 
That is, the largest error of e will occur at the time t — 0, or t = 7r,. 
or t = cos -1 If it occur at the time t — 0, its value 
w \ rl p 2 J 
will be 
p + l-r + v = t l (say); 
if at the time t = tt, its value will be 
— p + i + r + v = e 2 (say) ; 
/l 2 + r 2 rl\ 
and if at the time t = cos 1 A( — — - — ) its value will be, 
ri p2 
rl l 2 + r 
2p 
2 ^- P + v = H (say). 
• (io> 
(11) 
( 12 ). 
value given by (12), namely, t — cos 1 J 
Which of these three is greatest depends on the values assigned 
top and v; and our object is of course so to assign the values (i.e. 
determine the comparison S.H.M.) that for the chosen values the 
greatest of the three functions (10), (11), and (12) — whichever that 
turns out to be — shall be as small as possible. 
Before proceeding to this it should be pointed out that for 
values of p<rl/(l + r) or >rl/(l-r) the time at which e has the 
'l 2 + r 2 rl\ . . 
NT ~f 2 f 18 lmagmary; 
and therefore for such values of p (12), although itself real, has no 
physical meaning, and is to be ignored. 
It will be convenient to regard each of the functions (10), (11), 
and (12) as being a surface, determined by the two independent 
variables p and v. Taking then three mutually perpendicular 
axes forp, v, and e respectively, we have a system of three surfaces, 
consisting of two planes and an hyperbolic cylinder. Fig. 10 
represents the section of these surfaces by the p - e plane.* If any 
other section be taken parallel to the p - e plane, and at a distance 
- v from it, then, because of the manner in which v enters into 
the equations, fig. 10 will equally represent this other section, 
* The termination of the hyperbolic branch at its points of contact with the 
two straight lines, i.e., at the points the abscissae of which are respectively 
2^— and -2l— is simply because, as explained above, (12) is physically 
meaningless beyond these limits. As negative values of p are also meaningless 
in the present connection, all that lies on the left of the € axis has been omitted 
from the figure. 
