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DR. W. F. SHEPPARD ON 
(iv.) When j is relatively large, the solution given in (ii.) above can only be 
regarded as a formal one, since it involves calculation of determinants. I have not 
been able to provide a general solution which shall avoid determinants; but it will 
be seen in §§17-19 that, if we can find the values of certain quantities occurring in 
the formulae, we can deduce the X’s and thence the coefficients of the <x’s. These 
latter are important as giving us formulae which contain only a few terms and are 
therefore suited for numerical calculation. 
10. Expressions in terms of a Related Set .—Suppose that there is another set of 
l +1 quantities u 0 , u lf u 2 , ... u h connected with the cTs by linear relations; and let the 
set conjugate to the u s be y () , y u y 2 , ... y t . We shall take the relations between the 
us and the S’s and between the y s and the S’s to be, as in § 4, (r = 0, 1, 2, ... 1) 
U r = ( r o) <b+ ( r i) <h+ (r 2 ) <b+ ■•• + Of S h .(47) 
Hr — M <b + [^i] + [r 3 ] 4 + • • • + [r J S, .(48) 
(i.) Let the improved value of u r , using <7s after be v r . 
(XIII.), remembering that, by (36), 
i e f)j = o if / > y, 
we have 
V r = (n) (4 + ( r i) Mj + • • ■ + (■ r l ) ( e l)j 
= (n) ( e o) ; - + (n) (*i)j + ••• + i r j) i e j)j- 
Then, from (47), by 
• (49) 
. (49a) 
Thus the vs are related to the e’s in the same way that the us are related to 
the S’ s. 
(ii.) Similarly, if the improved value of y r , using <fs after Sj, is z r , we have 
^ = [n] («o )j + [n] Mj +... + M (ef .(50) 
= [ r o] {e 0 )j + [n] Mj + • • • + [vj Mj ;.(50a) 
and the z s are related to the e’s in the same way that the y s are related to the d’s. 
(iii. ) Let w be any l.c. of the S’s or of the u’s or y s, and let x be its improved 
value, using els after <5). Suppose that x is expressed in terms of the y s, the 
coeificients being p 0 , p u p 2 , ... p h so that 
and let 
so that 
x = PoV 0 +piVi +p 2 y2 + • •. + PiVi ; 
\ g = m.p.e. of x and (e g )j, 
\ = 0 if g > j. 
Then, by the condition of conjugacy of the w’s and the y s, 
(51) 
m.p.e. of x and u r — p r . 
(52) 
