THE STROBOSCOPIC EFFECT 
227 
the calculus. We are at liberty, however, to treat the simple 
stroboscopic image as an object, say a small circular dot (geo- 
metrical point), having continuous existence. Further, there 
can be assigned to it any velocity, as a continuous function of 
the time, that is consistent with the observed behaviour of the 
image itself. Call the geometrical point that represents the 
image the “index point,” and write, 
As = J D t s dt (16) 
ti 
whereDtS is the instantaneous velocity of the index point, a vec- 
tor function, and (t 2 — t x ) is the period of the stroboscopic image. 
Evidently Dts may be any function of the time whose integral 
between the limits t 17 t 2 , will give a 1 s. 
The most general treatment of the stroboscopic velocity is for 
three dimensions. Let the stroboscopic figures have a known 
distribution along a space curve, which may be called the curve 
of location. This curve is attached to the stroboscopic screen 
and moves with it. Let it be defined in terms of cartesian co- 
ordinates fixed on the screen (moving axes), and express the 
motion of the index point by the time functions, 
x'=f(t) 
y'=g(t) (17) 
zHh(t) 
These functions must be such that at any given “participating” 
illumination (“participating” in any simple image) the index 
point is at one of the point stroboscopic figures on the curve of 
location. Elimination of t from equations (17) will leave two 
independent relations, 
F(x, y'. /.')—() 
G(x', y, z')=0, (18) 
which represent two geometrical surfaces whose line of intersec- 
tion is the curve of location. 
Next refer the moving axes to a set of cartesian axes fixed in 
position. The transformation equations are, 
x= l t x'+ 1 2 y'-f 1 3 z'+h 
y— rn, x +m 2 y'+m 3 z'+j 
z— n t x'+ n 2 y + n 3 z'+k 
( 19 ) 
