ial fraction 
which is 
To combine the two terms, we multiply the numerator and denominator of the 
first by 1 4- x 2 4- x A , thereby reducing its denominator to 1 — ¿c 4 .1 — x*. 1 — a 2 . 1 — ft 4 , the 
denominator of the second term; then the sum of the numerators is found to be 
= 1 — a 1 2 
4- x (a 2 — a 4 ) 
+ X s (ft — ft 5 ) 
4- XT’ (ft — ft 5 ) 
4- x* (ft 2 — ft 4 ) 
+ x 8 (1 -ft 2 ), 
viz. this is — (1 — a 2 ) f 1 
4- o?x~ 
+ (ft + ft 3 ) X 3 
+ (ft + ft 3 ) X' 
+ a 2 x? 
Hence we have 
^ (■' ^ 1 — aa?. 1 — ax. 1 — ax~ l . 1 — ax~* 
4- 
4- 
1 — a? 4 .1 — of 1 — am? 
1 
1 — x 2 .1 — x 4 1 — ax 
1 + x* + (ar ! + x*) ft 4- (x 2 + X s ) ft 2 -f (x? 4- a 5 ) a 3 1 
1 — ¿c 4 . 1 — X* 
] -ft 4 ’ 
1 CLOG “f - 0?CG^ 
which is, in fact, the expression for = —= _ 4 decomposed into partial 
X CLOG • X CLOG • X “■ CL