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PROFESSOR A. R. FORSYTH ON 
E' 0 A' F" 
D' E' F' G" 
0 C B 0 
0 0 0 A 
(As the object is to illustrate the general case and not merely to get the result in 
this particular case we have not selected those rows which leave the denominator in 
the simplest form.) In the determinant of 15 2 constituents multiply each column by 
the quantity which stands at the head of it, add the results horizontally along all the 
rows, and replace the constituents of the last column by these new constituents which 
are, in order, * 
2 / 3 F l5 y~zF x , yz*F 1} z 3 F l5 z 2 F 1? y¥ v zF v F v y~ F 2 , yzF», z 2 F 2 , yF e , F 2 , zF 3 , F 3 , 
so that if we expand we have the numerator of our eliminant in the form 
A .r,]Fl + A .r, 3 F 3 + A .r i3 F 3 
where the A’s are determinants differing from the initial determinants in the last 
O 
column alone ; Ab tl has for its constituents there the coefficients of F 2 so long as Fj 
occurs in the later form and then zeros; A '* )2 those of F 2 where it occurs and else¬ 
where zeros; Ak , 3 those of F 3 where it occurs and elsewhere zeros. Moreover, we 
know that our eliminant is an integral function of x not extending in an infinite 
series; hence each of the coefficients A' must be divisible by u. If not, one of the 
F’s (say F x ) must be so divisible ; since u is a function of x only it follows that, when 
u — 0, F, = 0 whatever z and y may be. We shall assume that such factors are removed 
before the investigation begins as they are useless for the purposes for which the 
functions are required; and hence we obtain our eliminant in the form 
X= A^F 1 + A* )2 F 3 -b A* )3 F 3 . 
Similar remarks of course apply to Y and Z, the eliminants with regard to z and x, 
y and x. 
4. We may also obtain the result as follows : 
Between F^=0, F „=0 eliminate z and denote the eliminant by ; then, as we have 
already seen, can be expressed in the form 
X i/ =X„ 
t ~b k.F 
■p 1 - )>• 
Between F„= 0 , F^^O eliminate z and denote this eliminant by Xj,; then 
X' 2/ =fi u F n J ry r F'p- 
Now X ;/ , X' y are both functions of x and y ; eliminating y between them and 
denoting the eliminant by X we have 
