ON ELIMINATION IN THE CASE OF EQUALITY OF FRACTIONS. 



or 



i tt iAy 3 1 - { I fli&rs I + 1 <*Ay 3 I + 1 a i/ 3 2 c s I }»■ + {I a A c 3 ! + 1 «i/V 3 1 + 1 a i h -irz I } ? ' 2 - 



I a 2^374 ! - { I a 2/ 3 374 ! + I a -hli \ + I a A C 4 \} r + { ! a A C i ! + I «2& C 4 I + I «2^74 I V' ~ 



l^ 4 7il - { I «3&7i I + I a 3^yi I + I a 3& c i I ) r + { I a 3 Vi I + I ^AVa I + I «3 & 47i I }'' 2 - 



! a 4^i72 I - { ! «4&7« I + I a 4 & l7 2 I + I a 4^1 C 2 I }■'" + { I a A C 2 I + I a £l% I + I «4 6 l72 I }** ~ 



«A C 3 I r3 = ° 



a 2^'A C 4 \l' 3 = 



a z l> i c l I J' 3 = 



a 4^1 C 2 | ? ' 3 = t) 



and thus by eliminating r, r 2 , r 3 reach the result 



aj&yg | 2 | «i/3. 2 7 3 | 2 | a^^ | 



o-S-iYi I 2 | a 2/ 8 3 y 4 | 2 | a 2 6 3 c 4 



I *tPtfi | 2 | a^y! j 2 | a 3 & 4 c i 



I H^\/2 ! 2 | «4/8i7 2 I 2 I * 4 V 2 



«A C 3 I 

 I «2 & 3''4 I 



I a 3 i 4 Ci I 



I ^^jC, I 



= 



The general theorem is : — The eliminant of the set of equations 



a i X \ + \ X 2 + • ■ • + h X n 

 a i X l + P\ X 2 + • • • + \%n 



a n +iX l + h„+yc. 2 + . . . + l ll+1 x„ 



a n+l x i + Pn+l x 2 + ■ ■ . +K 1 x „ 



IS 



D x 2D\ 2D"j 

 D 2 2D' 2 2D" 2 



D„ +1 2D'„ +1 2D" )H 



(II) 



where 



D, 



QSj^o . . . ?„ 



1) 2 = I a 2 /; 3 



«„+A 



cmc£ where D f T indicates that any one of the letters D r has been replaced by the 

 corresponding letter of the other alphabet, D" r that any two letters have been similarly 

 treated, and so on. 



Of course, since the numerators and denominators of the given fractions may 

 legitimately be interchanged, it would be equally correct to begin in the first column 

 of (II) with a , ft , 7 , . . . instead of a , b , c , . . . , denoting the elements of the 

 column by \ , A 2 , . . . , and substituting A for D throughout the other columns. By 

 doing so, however, we should only be obtaining the same columns in reverse order, the 

 eliminant being expressible with a superfluity of notation in the form 



D, 



2D', 



D 2 2D', 



1), 



2D', 



2D", 

 2D" 2 

 2D"., 



D„ +1 2D'„ +1 2A" n+] 



2A' a A, 

 2A' 2 A 2 

 2A' A 



2D' )1+1 A B+1 



(4) In connection with the same problem, viz. the elimination of x , y , z when four 

 equivalent fractions are given, let us now consider the determinant 



