TRANSACTIONS. 



I, — Elimination in the case of Equality of Fractions whose Numerators and 

 Denominators are Linear Functions of the Variables. By Thomas Muir, LL.D. 



(MS. received November 6, 1905. Read same date. Issued separately January 6, 1906.) 



(1) It is well known that if equations of the type referred to in the title be dealt 

 with like ordinary quadrics, the eliminant obtained is marred by association with an 

 irrelevant factor. Thus, to take the simplest case, viz. 



a x x + b x y _ ii, 2 x + b 2 y _ a 3 x + b 3 y 



or 



we obtain 



a a a; + fi-g/ a 2 x + fd 2 y a. A x + /3. 3 y ' 



| a x a 2 1 x 1 + { | aJ3 2 i + | V, | }xy + \ h^ 2 \ if = 

 | a 2 a s j x 2 + { | a % B s \ + | b 2 a 3 1 }xy + \ b 2 /3 s \ y 2 = 



a x a 2 1 I aj/3 2 1 + | 5 1 a 2 1 | 6 iy 8 2 1 



| a x a 2 1 I a a /3 2 | + | 6 1 a 2 1 | b l fi 2 



i «2 a 3 I I a 2^3 I + I 6 2 a 3 I I b A I 



a,a., 



,jSo I + I &,a„ 



a,w 3 1 + I o 2 a 3 1 i b 2 f3 3 1 



= 0, 



the left-hand member of which contains the irrelevant factor | a 2 (3 2 1 , being readily 

 shown to be equal to 



a 2 f3 2 

 a. y b 



a-, O o 0U 



«l a 2& 



I 6 l a 2Ai 



The object of the present short paper is to draw attention to other modes of 

 procedure, and to formulate the results for n variables. 



(2) In the first place, then, it has to be noted that when the number of fractions 

 is the same as the number of unknowns, it is possible to express each of them in terms 

 of the coefficients alone. Thus, having given 



a x x + b x p + c^z _ a 2 x + b 2 y + c 2 z _ a 3 x + b 3 y + c 3 z 



a i x + P-& + yf- °-2 x + P$ + 72 z ~ a 3 x + P 3 y + y& ' 



and denoting each of the fractions by 1/r, we can deduce an equation containing only 

 r and the eighteen coefficients of x, y, z. To this end consider the determinant 

 TEANS. EOY. SOC. EDIN., VOL. XLV. PART I. (NO. I). 1 



