K. Pearson 267 
The present paper endeavours to indicate a systematic method for fitting 
curves. It is not claimed for it : 
(i) That it will succeed in giving the values of the constants in every 
conceivable case. 
I can only say that after an experience of some eight years' use by my fellow- 
workers, students and m} self we have found it applicable to a vast range of physical 
and statistical data. 
(ii) That it will give absolutely the " best " values of the constants in all 
cases. 
I endeavour to show that it must give good values. The definition of " best 
fit " is more or less arbitrary, and for practical purposes, I have found that with 
due precautions as to quadrature, it gives, when one can make a comparison, 
sensibly as good results as the method of least squares. 
Finally it is an advantage to have a systematic method of approaching curve- 
fitting problems, which at any rate gives practically excellent values for the 
constants in a very great number of cases in which the method of least squares is 
admittedly of no service at all. 
(1) General Theorem. 
A sei'ies of measurements or observations of a variable y having been made, corre- 
sponding to a sei'ies of values of a second variable x, it is required to determine a 
good method of fitting a theoretical or empirical curve y = cf) {x, Cj, Cg, c^, ... Cn), where 
Ci, Ca, C3, ... c,i are arbitrary constants, to the observations for a given range 21 of the 
variable x. 
Such problems in curve-fitting recur with great frequency in physical, 
biological and statistical investigations. The usual theoretical rule is to use the 
method of least squares, but if the constants Cj, c.,, c^, ... Cn are involved in a 
complex manner the equations obtained by the method of least squares become 
unmanageable, and we find physicist and statistician remarking that " the increased 
accuracy of the result obtainable by least squares would not be an adequate return 
for the labour involved," and then falling back on some more or less questionable 
process of determining the constants. This process may be graphical or arithmetical, 
but it is usually unsystematic in character and elastic in result. The object of the 
present paper is to give a systematic method of fitting curves to observations, 
which I have reasonable ground for considering a good one, and which at any rate 
for a great variety of problems leads us to easy and simple results. 
The assumption to be made solely for the proof, but not in practice, is that 
y = (f){x) can be expanded by Maclaurin's Theorem, and that the resulting series 
converges fairly rapidly. Let the expansion be : 
