300 
Miscellanea 
We can now by aid of (v) express rjy^^. We have : 
i-y, . = + f 12V^2 + (*13 - |j f 12) y (^4 - <^3-7<^>2) 
, f"-l'""£('"7:'")/*-^-*'r ,... 
<^.0-<^4'^/<|)2-^y(<^4-</)3W 
The conditions therefore for linear regression, or = are : 
fi2=fi3 = fi4 = etc.=0 
That is : ^10 = r s/^^, q^^ = r/Sj , = »• A , etc 
V/3i 
For parabolic regression : ^,3 = ^ fi4 = ^ fw, etc , 
or ■ ^,3 _ ^3 ^ j^^^ _ ^_3 
And lastly for cubical regression : 
•"-i'-*^'(-:i<»)/(*.-i>»-- ■■ . 
Such conditions, especially with regard to their proliable errors, become less and less manageable 
as we proceed. 
The general principle involved in the present paper has been discussed hy Tchebycheff *, and 
more adequately by J. P. Gram f, but the former had in view the fitting or graduating of curves. 
He calculated quantities which correspond to our fig's on the assumption that 71^=1, i.e. that the 
weight of the ^^'s are all the same or that the marginal total is a rectangle. He was thinking of 
fitting a curve to a curve and not fitting a curve to a swarm of points. In his case each fi^ and 
accordingly each /3 and each ^ is expressible in terms of the total number m of subranges which 
he takes of equal length. There are I think simpler methods of calculating the equation to a 
higher order- parabola in such cases |. As far as I am aware these orthogonal regression functions 
have not hitherto been dealt with and they throw a good deal of light on the original equations 
I provided in 1905 for skew regression. I had not recognised at that time that my expressions 
of each order were true orthogonal functions. It will be seen that my solution does not involve 
equality of subranges and is not limited to any S[)ecial frequency distribution. 
II. Note on the "Fundamental Problem of Practical Statistics." 
(Biometrika, Vol. xili, p. 1.) 
Some misunderstanding has arisen with regard to my paper under the above title in the last 
issue of this Journal. I believe it to be due to the critics not having read Bayes' original theorem 
as given by Price in the I'/dl. Trans., Vol. liii. Bayes takes a ball and places it at random on a 
table, say of breadth unity, and its distance from one side being x, its chance of falling between 
X and .r + S.r is S.r. x \s tluis not a chance, but a variate. He now calls a " success," the chance that 
any other ball placed at random on the table will be nearer to the same side than the first 
* Mimoires de VAcadimie de Saint-Pitersbourg . Memoiis in 1854 and 1859. A risumi by R. Badau : 
Bulletin Astronomique , T. viii, Paris, 1891, pp. 350, 376 et seq. See also Liouville's Journal, 2' S6rie, 
T. Ill (1858), p. 289 et seq. 
t Thesis: " Om Raekkeudviklinger bestemte ved Hjaelp af de mindste Kradvaters Methode." Kj£(ben- 
havn, 1879. 
X Biometrika, Vol. 11, pp. 12 — 16. 
