^66 L. EULERI OPERA POSTHUMA. Anaiym. 



Valores ergo y et p arbitrio nostro relinquuntur, atque alterum quidera sine ulla restrictione ad 

 lubitum determinare licet. Ponatur ergo yy=zA et /) = cc, fietque 



a = — ccl/^, r = y^, ^ = y(^-t-Ccc-+-£;c*) et ^ = ^ 

 et aequatio canonica hanc induet formam „ -. 



= — Acc -*- A {xx -+- yj) h- 2(cyy(A -+- Ccc -+- Ec^) A — Eccxxyy. 



IIL 



Antequam autem his litteris majusculis utamur, differentiemus ipsam aequationem propositam 

 dx{yx-i- dy-+- ^xyy) -\- dy (yy -¥-dx-^ ^xxy) = 0, ^® •**<^''* 



quae abit in hanc ^ = 7 ^» 



yy-+-ox-\-^.rxy yx-t-8y-t-^scyy 



Substituendo ergo loco horum deno^iinatorum valores surdos primo inventos, habebimus per Vp 

 multiplicando 



dx dy 



V{A -H Cxx -i- Ex*) V(A -¥-Cyy-\- Ey*) ' 



>tUHUIt> itlHi 



IV. 



,^j^^^^,Eroposita ergo hac aequatione differentiali 



... dx dy 



(niiJuo^ — 



V(A-^Cxx-t-Ex*) V{A-*-Cyy-i-Ey*) 



ejus aequatio integralis erit 



= — Acc-\-A(xx-^yy)-\-2xyy(A-i-Ccc-\-Ec^)A — Eccxxyy, 

 quae cum constantem novam c ab arbitrio nostro pendentem involvat, erit adeo integralis completa. 

 Inde autem oritur 



I — X V{A -i- Ccc -t- £c^)y ±cV{A-i- Cxx -t- Ex*) A 



^ A — Eccxx 



ubi quidem signa radicalium pro lubitu mutare licet. 



luieiTKf^ 'y 



Cum igitur posita nostra aequatione canonica sit 



JV{A-¥-Cxx-\-Ex^) Jy{A^Cyy-*-Ey*) 



ponamus ad alias integrationes erueudas 



r ii^xdx y y ydy y 



JV {A -H Cxx -t- ,£«*) JV{A -*-Cyy-¥- Ey^) ~~ * 



erit ergo loco radicalium valores praocedentes restituendo 



xxdx yydy dV 



yy-*-dx-t-^xxy yx-t-Sy-t-^xyy Vp 



bincque porro r j: 



xxdx [yx -+- dy-i- ^xyy) -t- yydy (yy -i- dx-i- ^xxy) = 



y (y 8 {xx -i- yy) -*- (yy-t-dd^xy-t-^^x^y^-t-y^xy {xx-k-yy) -*-2d^xxyy), 

 QC 



