INDETERMINATE PROBLEMS. 205 



By reducing the firft equation, x zz - s — — J and by reducing 



1 r- 1 y+d — m h ■ m V — md — f y-Vd — mh 



the iecond, x zz • whence — — , and con- 



' mg ' e mg ' 



fequently,/ = -*****&+ *-** . But * zz J2=2±£, there- 



* J J m z g — e e ' 



r . r 2md — m z h+f 



fore alio x zz : • 



m g — e 



Suppose 14^+3 ix-f-24 zz y 1 ; then, taking 9 — d 2 from both 

 fides, i4* 2 -i-3iH-i5 = y 2 — d 2 ; but ^{b % — ^ac) zz\/(g6i — 840) 

 zz it $ whence, if n zz 2, the divifors, by Cafe III. will be 

 7 X ~T" 5 an d 2A7+3* wherefore, making w zz 2, x zz 



r -^l = 5 ; andj , - 2 4 +2 o4-,r- 42 = ^ For 4.25+31.5+24 



= 529 = ( 2 3) a - 



PROBLEM VIII. 



Let c and d be known values of x and y in the exprejfion^ ax 2 -\-b 

 zz y 2 , and from thefe let it be required to difcover others. 



Since ax 2 -\-b zz j/% and ac 2 -\-b zz d% fubtracling thefe equa- 

 tions, we fhall obtain ax 2 — ac 2 zz y 2 — d 2 , and by refolution, 

 [ax — ac)(x-\-c) zz [y-\-d)(y — d) ; whence ax — ac zz my—md r 



and x-\-c zz — — . From the firft of thefe equations, x zz 



my — md+ac y+d — mc my — md+ac 



, and from the iecond, x zz : ; whence 



y+d—mc_ , m*d+ad—2mac ^ M (m 1 +a)d—2mac r> 



zz — - — -, and y zz £ ~, or — i — -^ — . But x 



m — a '- mr — a 



