138 DE CRITERIIS INTEGRABIL. 



U-rz=(t^)4-;.(t^)4-^(l5.)_H,(i^) etc. 



nam ob y zr o , crit 



(^)-(^) — oi (^) — (^)=:o- (^-:^)=i(^-^)==:oetc. 



7. Progretiiamur riunc ad demonftrationem in- 

 lignis iftius Theorematis , quo ftatuitur formulam 

 V d X fore integrabilem , fi poruo 



dWz:zy[dx-i-'N^y-\-'Pdp-{-'^ciq-{',..-\-\]du fuerit 



d? ddQ d'K d^^U 



dx d X d X — d x^ 



Et primum quiJ.em ex aequatione propofita , opb 

 integratioiium dcducitur 



lioc autem vjlore ipfius U fubfiituto in aequationc 



.'^ V := M ^.v-1- N dy-^-Vdp -\-\}du , orietur 



i\I-m.i'\'^dy'\-Vdp^qdq...,-\-Tdt 



^duif^^^^Ndx-f ^"^-'^Vdx-^-f^^-^^Qclx,,. ^fTdx 

 Hinc vero per transformationes in § 5 all.itas obtinetur 

 \dxzzdxfMdx-i-dyfl<ldx-{-dp{fVdx-f^'~^Ndx) 

 Jfdq{f(idx-f^''?dx+f^'^Ndx)... 



4-^^( /Td^;»;. . ± / ^'"-'Wjc q=/ ('") W.v j. 



Si iam Tt antca fupponamus 



\dx=ziJidx^vdy-\~i:dp-\-Kdq,...-\-rdt., habcbimiis 

 IK-f^dx; v-/Ndx; 'K-fPdx-/^''Ndx-/{?^y^dx 

 K=zf(ldx-'J^^Fdx-\-f^'^Ndxzzf{(l-'Ti)dxcK. 



Pcno 



