92 
MR. G. J. BURCH ON THE TIME-RELATIONS OF THE 
This change necessitated a reconsideration of the method of analysis. The curve is 
a variety of the logarithmic spiral, but the intercept between the tangent and the radius 
vector on the asymptotic circle—the line corresponding to that which I had employed 
in the case of the rectilinear motion, is not constant, and not of a form convenient to 
use. There is, however, another property of polar coordinates which is at once 
available; namely, that the expression for the polar sub-normal is 
dr 
In this case 
dr dy 
To ~ ~~ d~0 ~ cy ’ 
so that this property enabled me to analyze the curves with remarkable ease, and I 
propose to abandon rectilinear motion for the plates in future.* 
Fig. 3. 
~7 
Fig. 4. 
The problem of the analysis of a photographed excursion resolves itself therefore 
into the determination of the length of the subnormals of a sufficient number of 
points upon the curve, together with the measurement of the distances of these points 
from the circle corresponding to the position of the meniscus for zero potential. 
These measurements are most easily made upon the instrument represented in fig. 3. 
The negative is fixed to a carrier B, pivoted at 0, on the base-board A, in which a 
* See Williamson’s ‘ Differential Calculus,’ 7th edition, p. 223, from which I got the clue while trying 
to elaborate a roundabout method of reproducing the polar curve in rectangular coordinates, so that I 
might analyze it. 
