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IOWA ACADEMY OF SCIENCE Vol. XXIV, 1917 
by (as)* the distributional displacement at right angles to the 
film motion. For the component in. the film direction, 
t e t a 
(As) y =j v dt - j Do dt ( 14 ) 
tl tx 
Thus, 
As= ^(As) x 2 + [/ v dt - j Do dt ]" (i^») 
tl t'x , 
The average instantaneous velocity over the common periods 
of the elementary simple images is obtained by dividing A s in 
(15) by (t 2 — tx), the interval between two flashes on the movie 
screen. For simplicity the film velocity v may be taken as con- 
stant, and the first term in (14) becomes v/B, where B is the 
number of projections per second. To get the stroboscopic veloc- 
ity on the movie screen, v s , obtained from (15), must be multi- 
plied by the magnification. 
It may be convenient to consider the distributional displace- 
ment in the y-direction, of the elementary figures. In this event 
let Dx be the distance between centers of successive pictures on 
the film, then D 0 =Dx-f- A y, which is to be substituted in (15). 
For the stationary condition lengthwise of the film we have 
A=B, and for the rectangular direction, A— 0. For motion in 
the latter direction A may still be regarded as zero and only 
(As) x considered. If v s is to be expressed in terms of A then 
when y s =0 the A for either direction may be taken as any 
value or any function of the time consistent with the observed 
stroboscopic motion. 
Many stroboscopic devices (toys) are similar to motion pic- 
tures, particularly in having n/m=l. 
IV. A General Mathematical Treatment of the Simple Stroboscopic 
Image. 
Stroboscopic velocity cannot in general be treated as an ordi- 
nary velocity for it deals with a moving object which, in its 
physical phase, is discontinuous. We can divide a definite dis- 
placement a s by a definite time 1/B and obtain a quantity that 
has the nature of an average velocity, but beyond the interval 
of time 1 /B we cannot go to carry the ratio to the limit as in 
