112 ON THE EQUATION OF DIFFEEENCES FOE AN EQUATION OF ANT OEDEE. 
which is equal to 
— Vlah 
+ ( — 1 2ac — + ss' +5^^) 
+(36ac— 72J^+9a^^)ss' 
+(2«-725c+12«J^)(s+s') 
_ +96^6?— 144c^+( — 48«(?+725^)^— _ 
+4(3a5®+12fe+12c+«^)(3«5'^+125s'+12c+«^) ; 
or reducing and dividing by 32, this is 
{9(ac } ss' 
{9(ad—bc)-{-Qab6}(s-\-s') 
-\-3Q{bd—c^)-\-(—15ac-\~27b^)0—a^d‘^, 
the terms in s’^ disappearing, as they should do. Writing this under the form 
( 9{aG—b^)-{-2>a^6^ 9{ad—bc)-\-Qab6 *^5, 1)(5', 1) 
I 9{ad—bG)-\-Qabd, 2)^{bd—G^)-\-{—lbaG-\-2'lb^)d—a^0^ | 
and equating the determinant to zero, we have the required equation in ^ ; the fonn is 
precisely that which is obtained by the ordinary process of applying Bezout’s method to 
the two equations 
(3«, 12^, 12c+ l)-^=0, 
(35, 12c+4«^, 12d-\-mj^s,lY=9, 
being in fact the before-mentioned equation 
{a^&^-\-{\baG—21b^)d— ?>^{bd—G^'fj{a^6-\-o{aG—b'^fj-\-o{2ab6-\-o{ad—bG)Y=^9. 
But, as already remarked, this elimination process is less convenient for the complete 
development of the result, than the method first explained in the present memoir. 
