228 
IOWA ACADEMY OF SCIENCE Vol. XXIV, 1917 
where 1, m, n, are direction cosines, h, j, k, are coordinates of the 
origin of moving axes referred to the fixed axes, and x , y', z', 
are coordinates of any position of the index point referred to the 
moving axes, — all functions of the time. The direction cosines, 
which are not all independent, together with h, j, k, are known 
functions of the time for they describe the motion of the strobo- 
scopic screen. There follows, 
As =^(Ax) !j +(Ay) 2 4-(Az) 2 , (20) 
where 
t 2 
AX=j9X/3tdt 
ti 
and similar expressions for Ay and a i z. The partial derivatives 
give the three instantaneous components of the velocity of the 
index point. The displacements A x >Ay?Az, are those of the in- 
dex point with reference to the fixed axes, and they have oc- 
cured by motion of the index point along the curve of location 
during the time interval (t 2 — t x ). The stroboscopic velocity is, 
v s — ^ ( 9 x / 9 t) 2 + ( 0 y / d \) 2 + ( 9 z/ d t) 2 , (^1) 
where the values of the partial derivatives may be obtained 
from (19). 
For the stationary stroboscopic condition we have, 
li x + 1 2 y +. la z 
m, x'+m 2 y'+m 3 z'+j==C 2 (22) 
n x x'+ n 2 y'+ n 3 z'+k=C 3 , 
where the C ’s are constants. 
For the tonoscope and the tonodeik, if the tangent plane of 
the drum is taken as the stroboscopic screen, the conditions are, 
1 ,=!, 4 = 1 3 =0 
m 2 =l, m 3 =m 1 =0 (23) 
n 3 =l, n x = n 2 =0 
k=0, 
provided that the plane stroboscopic screen moves in its own 
plane without rotation and along the y-axis, and that the axis 
of x' is parallel to the axis of x, the axes of y' and y lie in the 
