518 
Proceedings of Royal Society of .Edinburgh. [sess. 
The solutions are as follows : — 
If b = c = 0, then 
_xf 
y = y 0 e * 
where y 0 is the ordinate at the origin. This is the normal proba- 
bility curve discovered by Laplace and Gauss. 
If the roots of a + bx 4- cx 2 are real, 
Type I. y = yll + £.YY 1 - where -i = 
\ / V CLq / Mq 0*2 
If both roots he real and equal, 
Type II. 
2/ = 2/o( 1 
If one be infinite, the solution is 
Type III. 
f xX 1 * 
y=V \ l+ a) ^ 
If the roots are imaginary, one solution is 
-i x 
v tan 
Type IV. 
y=y o 
(•♦sr 
This last solution is the one which is found to he a good inter- 
polation curve for epidemics. 
Types V. and VI. do not concern this paper. 
The method of fitting statistics to these curves is thoroughly 
described in a paper by Professor Pearson in Biometrika , vol. i. 
and vol. ii. part 1, entitled “ On the Systematic Fitting of Curves 
to Observations and Measurements.” 
The theory by which the curves are derived is fully discussed 
in the same journal, vol. iv. parts 1 and 2, in a paper written, 
justifying his methods, by Professor Pearson, entitled “ Das Fehler- 
gesetz und seine Verallgemeinerungen durch Fechner und Pearson : 
A Rejoinder.” The subject was originally developed in two 
papers in the Philosophical Transactions of the Royal Society , and 
is best read in these papers. The references are “Contributions 
to the Mathematical Theory of Evolution. II. Skew Variation in 
Homogeneous Material,” Phil. Trans., 1895, vol. 186a, page 343; 
and “X. Supplement to a Memoir on Skew Variations,” Phil. 
Trans., 1901, vol. 197a, page 443. 
