218 
DR. W. F. SHEPPARD ON 
We should have to choose the e’s so as to make 
227r r , s (y r -z r ) (y t -z t ) 
r s 
a minimum, where 
TT r<<i = rn.p.e. of u r and u s .(64) 
This would give 
2 ( r /) { Z r-Vr) = 0.(65) 
r 
(iii.) The e’s given hy (65) are the same as are given by (61). For we have seen in 
§ 4 that 
(r f ) = rn.p.e. of u r and ay,.(66) 
[sy] = rn.p.e. of y s and ay.(67) 
If we express the u s in terms of the <5’s, and write 
2 [s f ] u s = ZA t S t , 
S t 
then, by the condition of conjugacy of the o-’s and the 8’&, and by (66), 
A, = rn.p.e. of <r t and 2 [sy] u s 
S 
= 2 [«,] («,)• 
This is symmetrical, and we should get the same expression for the coefficient of S t 
in 2 (?y) y r , so that 
2 0 /] u s = 2 (r f ) y r .( 68 ) 
S O' 
Similarly, if we substitute the values of v s from (57) and of z r from (63) in 2 [sy] y. 
and in 2 (ry) z r , the coefficients of e t in the resulting expressions are equal. Hence 
r 
(61) and (65) are identical equations in the e’s. 
(iv.) The values of the e’s as given by these equations are in fact independent of 
the us or the ys. For the value of A t as found in (iii.) above is 
2 [»/] ( s «) = m.p.e. of 2 [>y] u s and 2 (s 4 ) y, 
s .s s 
— rn.p.e. of ay and <r t , ' .(69) 
hy (24) and (16). Hence, denoting the rn.p.e. of ay and cr t , as in §3, hy the e’s 
given hy ( 6 l) or (65) are the same as would be given by (f— 0, 1, 2, ...j) 
t=j t=i 
2 >]f, t e t. = 2 >1/, t 
t = 0 t = 0 
(70) 
