292 
PROFESSOR K. PEARSOX OX THE MATHE^LATICAL THEORY 
distinction between mean and mode,"^^ the proljabilities of negative and positive errors 
of the same magnitude Ijeing quite different, the abnormal concentration of errors 
round the mode, are all characters of the distribution, which must be taken into 
consideration, and vdiich it is important to describe. 
In concluding this jiajDer I desire to heartily thank those who have aided me in its 
preparatic n. In the first place iny gratitude is due to Dr. Lee, Mr. G. Lb Yule, 
and Dr. Macdoxell for the time and care they gave in experiment and observation. 
In the next place I owe Dr. Lee special thanks for the constant assistance I have 
received in the laborious computations she has aided me in, and which are hardlv 
obvious on the face of this paper. To Mr. K. Tressler I am indebted for great 
assistance in tire conduct of the bright-line experiments, especially in the preliminarv 
adjustments we had to go through before we got our apparatus into efficient rvorking 
order. He has also prepared from tlie calculations of Dr. Lee and myself the whole 
of the frequency diagrams. The work of experiment and reduction has extended 
over nearly six years, during which considerable progress has been made (c.p'., by 
Mr. Sheppard’s discovery of the l)est corrective terms for the moments) in statistical 
tlieory, and thus all our data have not been dealt with in an absolutely uniform 
manner ;t but the divergences due to method are small as compared with the 
probable errors, and we have taken great care by duplication of calculations to avoid 
as far as possible arithmetical blunders. 
* Soine American writers persist in taking the maximum group of ohserved frerpiency as the mode. 
But the fluctuations of random sampling make such a deteiinination of tlie mode in many cases cpiite 
futile: see for examples my Diagrams YL, IX., and XII. The mode is where dy^tlx vanishes for the 
iheoretical frequency curve, and is not visible on mere inspection of the ohseivations. 
t The calculations for the bisection of lines were in part made on the grouped observations without 
Sheppabd’s corrections, with the value of the mean error as given in the usual theory. 
