Cum igitur fit ^ rr V X -H ^' Y, aequatio integralis , quam 

 fumus adepti, erit 



yX-t-yYr=y(AH-2B (x~\-j) -+-C (a:+^}0» 

 quac adeo eft algebraica; vbi notetur efle 



X=:A-hBx-\-Cxx et Y — A + B/ + C j' j'. 



§. 29. Sumamus igitur quadrata, et noftra ae- 

 quatio integralis erit 



2 A + B (a- + j) -4- c (x* -^r) -h ft y X Y 



z=A+iB{x+j) -+- C{x H- j)% fiue 

 a A -B {x -f j) -aC.vj -t- ^yXY^A, 

 quae penitus ab irrationalitate liberata, pofito A — 2 A 2: F 

 praebcbit 



4XY=r4AA + 4ABfjr+j) + 4AC {xx-\-yy) 

 ■+^BBxj-\-^BCxy {x-^y) -^^CCxxjy 



~ r'+ 2rB(A- + r)+4rCA:j + BB (jt +^)' 



+ 4 B C A- j/ (JT +^) + 4 C C .V xjy 



fiue 



(^AA-r^^ + ^B^^A-r^^jr+j^^+^fBB-rCjJr/ 

 + 4AC(.vjf+^j') - B' (jr+^)' =0. 



§. 30. Quod fi iam hanc aequationem rationalem 

 cum formula canonica ^ qua olim fum vfus ad huiusmodt 

 integrationes expediendas, comparemus, quae erat 



a + 2 (3 (jc -{-y) -{■ y{xx -\-yy) + 2 5 jr j' = o , 

 dum fcilicct loco {x-\-yY fcribamus {xx-{-y y) + 2 Jf J, 

 repericmus forc 



« — 4AA-r»; (3 = B(2 A-niY^+AG-B*; 

 a^mBB-^rG. 



ASfaAvadlmp.Sc.Tom.lfP.L £ §• 3X« 



