Mr Wallace on ihe Elhmnaiioa 
Lc. — = t?, then w = ^ — ^-r\— > V = These values 
s + \ ^ -{•I 
of a? and y being now substituted in the second equations, it be- 
comes, after proper reduction, 
{a 2 h +2 (2 c — d) — -n® j* 
— 2bmv—n^ — a 
an equation of the fourth degree, by which v may be deter- 
mined, and thence the values of x and y. 
2, Let the equation, which is of the second degree, be 
x^-^axy-^hy ( 1 ) 
and let the other be ^ 
X* y^-\-K xy ^‘*' + &C.rr:0 (2) 
an equation of any order whatever. 
In this case, we must make 
_(r^ — h s^) c _{2 rs as^) c 
r^-\-ars-\-bs'^'^ -{■ars-\-b 
and then, independently of any particular values of r and 
we shall have axy -\-hy ; if we now make ^=1, 
(which comes to the same thing as to substitute 
we shall 
have X and y expressed by functions of r only ; and these being 
substituted for x and y in the second equation, the result will be 
an equation involving only v : this being resolved, will give 
and thence x and y. 
3. The most general equation of the second degree may be 
put under this form : 
x^~\-axy-\- by ^ -\-c x-^^dy—e. 
This, by the usual transformations taught in books on Alge- 
bra, may be changed to 
a7'^+A3/'^=B, 
where x and y denote certain functions of x and y^ which are 
at least rational in respect of these quantities,, although they 
may contain known irrational numbers. This equation will be 
satisfied, if we make 
2rs sjC 
