CHAIN OF CYCLOSALPA AFFINIS 409 
tion. Out of this examination has come the graphs shown in figs. 
4, 5, 6, 7, and 8. 
A curve was computed to fit the graph (fig. 2), as nearly as pos- 
sible. From the equation of this smooth curve we get a ‘calcu- 
lated value’ for each zooid; that is the length of each zooid, if the 
series were as smooth as our calculated curve. Wenext subtract 
the observed length of each zooid from the calculated length, and 
get a series of values, some plus and some minus according as the 
irregular graph went below or above the smooth curve. When we 
plot these plus and minus values above and below a horizontal line 
we have the graph fig. 7. It shows that the values follow the 
curve fairly well at first and then vary more and more; in other 
words, that we have a periodic curve of increasing amplitude.’ 
i Mr. McEwen gives the following summary of the method used: The sizes 
for each of the points corresponding to the numbers 45, 50, etc., to 90 were taken 
as the ordinates of a curve whose abscissae were 1, 2, etc., to10. It was assumed 
that the above curve corresponded to an equation of the form 
y=a-+bai+c2? 
and the most probable values of the coefficients a, b, and c were computed accord- 
ing to the method of least squares. By substituting (2x — 8) for az: in the above 
equation, the equation 
y=a+b (2e—8) +c x—8)? 
was obtained in which, if ;/y of the number of the point is substituted, will equal 
the computed value of the corresponding size. (This equation was used to calcu- 
late the corresponding values of y, which were used in connection with the observed 
values for computing the algebraic sum of the residuals and the probable error, 
for the purpose of determining if the equation was a proper expression for 
measured values of y.) 
It was assumed that this equation, determined from the 10 points was very 
nearly the same as if it had been computed from the 45 actual points, and there- 
fore represented the relation between the number and the average size of all the 
points. This assumption was verified in one case by including all the points and 
comparing with the result when only 10 points were used. 
The observed values of y were subtracted from the corresponding computed 
values and these differences were plotted as ordinates against the numbers as abs- 
cissae, thus giving a representation of the deviation of the observed values from 
those given by the equation. These deviations are due to errors in the measure- 
ments, and to the fact that the assumed equation was not a true expression for 
therelation. Asthe error inmeasurement was = 0.1, itis evident that the devia- 
tions are due mainly to the latter fact. 
The periodic character of these curves shows that the true law is a periodic 
fluctuation of increasing amplitude about a mean value increasing in a regular 
manner with the number of the point. 
