INDETERMINATE PROBLEMS. 199 



be nearly equal, the approximation will be more accurate. 

 Let m — a \ then the denominator m 2 — ab will be fmall, and 

 therefore the fractions large j whence, by fubftitution, 



. a 1 +/ib+2a z a*+ab+lab 



T,a-\-b : ^b-j-a, nearly. 



Thus ^49 : v/5 : : J 97 : J 99 '• '■ 7 '• 7+I^» whence v/5 ■= 

 7,07107, true to the laft place. 



Cor. 2. Let zb' = — — ; then m 2 — ab — (— J — aZ> rr (-^~) , 

 which, when 4 and b are nearly equal, will be fmall, and by 



( ) +ab+a(a+b) { 1 + a b+b(a+b) 



fubftitution, s /a • ^b : : ~ ^ : • " /a+bv , nearly | 



*{—)- ab \—)- ab 



hence, by proper reductions, y/a : Vb : : $a 2 -\-ioab-\-b 2 : 5/^-f- 

 ioab-\-a 2 . This formula is more intricate than the former, but 

 ftill more accurate. Thus, ,/9 : \/io : : 405+900-f-ioo : 500-f- 

 900+81 = 1405 : 148 1, and /io = 3,16209, true to the laft 

 place. 



PROBLEM VI. 



Let it be required to find a number, fuch that, if given multiples 

 of it be increafed by given numbers, the produB of the fums Jhall be 

 a fquare. 



Let (ex-\-f)[gx-\-ti) — y 2 ; by afTumption ex-\-f — my, and 



v 

 gx-\-h rr ~ Tranfpofing the firft equation, and dividing, 



m y f 



x ss j^t. — . Reducing the fecond, mgx-\-mh ■= y } and tranfpo- 

 fing 



