116 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
The quotient of the two determinants (A), when expanded by Schwein’s 
theorem, gives 
S«, 
%s 2 u s 
2s 4 «s 
2,S 6 ‘U s 
2s% s 
S 2 
S 4 
So 
s. 
S 2 
S 4 
So 
2w s 
%sH s 
S 2 
s 4 
So 
s 2 
S 4 
S 4 
So 
2s 
S 2 
S 4 
So 
S 8 
+ s 2 
S 0 
s 2 
-+ 
s 4 
S 6 
S 2 
S 4 
s 8 
S 6 
S 8 
2,0 
S 4 
So 
S 8 
s,„ 
+ S„ 
S 0 
s 2 
So 
S 2 
So 
S 2 
S 4 
So 
S 2 
S 4 
So 
S 2 
S 4 
So 
s 2 
S 4 
S 2 
s 4 
S 2 
S 4 
S 6 
s 2 
S 4 
So 
S 2 
S 4 
So 
S 8 
S 4 
So 
S« 
S 4 
S 6 
S 8 
S 4 
So 
S 8 
s,„ 
% 
So 
S 8 
Sio 
S 12 
a form more suitable for computation. 
If we take the first term only we obtain the graduation formula 
which would be obtained by fitting a straight line to each set of 2m + 1 
observations; the first two terms give graduation formulae obtained by 
fitting a parabola of the second or third order ; and, in general, from 
r terms we derive the graduation formulae obtained by fitting a parabola 
of the (2r— l) th or (2r— 2) th order to each successive set of 2m +1 
observations. 
The results of computations based on the above expansion are 
collected in the following table (Table I), which gives the coefficients 
M ,p 0 ,Pi ,p% ... in the least-square graduation formulae 
Mw 0 / =Po^O+J c 'l( W l + w -l)+P2( M 2 + “-2)+ * • • + 
formed on the assumptions k = 0 , 1 , 2 , 3 , 4 and for a considerable range 
of m.* The table, in fact, gives explicitly all the graduation-formulae of 
the least-square type which are ever likely to be used in practice. 
* I must express my obligations to Mr John Maclean, M.A., B.Sc., Professor of 
Mathematics in Wilson College, Bombay, and to Mr Jason J. Nassau, C.E., M.Sc., 
Instructor in Mathematics in Syracuse University, N.Y., U.S.A., who took a great share 
in performing the computations, which were carried out in the Mathematical Institute of 
Edinburgh University. It should also be mentioned that Mr W. F. Sheppard ( loc . cit.) had 
previously discussed the cases ft = 0,l,2; but his formulae, being expressed in terms of 
c 0 (Sheppard’s 6 0 ) , 9 2 c 0 , 9 4 c 0 ... in one paper and in terms of central sums in another 
paper, are not the same as the formulae found above under the headings k— 0,1,2, 
although, of course, they may be shown to be equivalent to them. 
