876 
MAJOR P. A. MACMAH0:N^ OX THE THEORY 
The relation 
N _ » / _^A^\ _ f 2^ - 2) - (2^ - 2) &,+ ■■■ + ( - )» (2» - 2) h, 
T» ~ 1 r 1 - Sj (1 - SJ (1 _ S,). ..(1 - SJ 
now becomes 
D=l + 
1 - S. 
. ” s„ s„ A^ A^, 
(1 - S..) (1 - S.J 
+ ... + 2!'a^ 
• S, 
• ^ • • • 
the final fraction 
1 K1K2 . . ■ 
(1-Si) (1-S2)...(1-S„) 
vanishing as will be seen presently. 
X 
48. This form of the fraction — will be employed in order to show that the expan 
sions of ^ and ^ are effectively identical. This arises from the circumstance that 
the right-hand side of the above identity when expanded contains no single term ex- 
])ressible as a function of '^“25 
49. 
Of the fractions X 
consider the typical fraction 
(2g<, + . ■. + -f + ■ ■ • + g,;) _ 
1 ~ 1 ”■ (2«i + ... -t- 2a^j -f- -f ... + «„) ’ 
since the numerator contains a factor and no quantity the fraction when 
expanded can contain no term which is a function of •52“2 j • • • alone. 
Consider next the typical fraction 
the numerator is 
and 
TkiK3 
(1 - S,d (1 ’ 
— 2a, 
A, A, = (2a -f., 
Kj Kj \ 1 I 
-j- 2a,^_i + ... + ... + a^,) (2ai-|- ... + 2a,^ fi- ... + 2a,^_i+ ... +a«), 
and, therefore, contains a term 
