quo facto aequatio illa propofita, femper fub hac forma reprae- 

 fentari potcrit Vdii — o, \bi V defignat functioncm quandam 

 ^uantitatum u, x,y, z, p,p' etc. y, #' ctc. r, r' etc. etc. porro 

 autem capto differentiali ipfius V, fit 



dV = M</«-f-N d x-t-K 1 dy -f-N" dz etc. 

 ■P <f j> H- V dp' -f- P" <//>" etc. 

 ■Q_dq-\-Qdqi-\-Q'dq" etc. etc. 

 formula haec differentialis femper crit integrabilis, fi fuerit 

 N — «L p -4- dd - a — d -i- R etc — o 



N' - J£ -+• "« - ^etc. — o 

 N« - ^ -f- 1 LfL^' - ^V- etc. — o 

 ctc. ctc. 



cuius proprietatis demonftrationem legerc licet, in DiiTertatio- 

 ne mea de criteriis integrabilitatis Tomo XV. Novor. Comm. 

 inferta. Propofitae nunc fmt hac aequationcs differentiales : 



ddx-\-a. d x -\- $ d y -\-y x -\-%y zzlo 



d dy +- a! d x +- (3' dy -f- y' x -f- h'y = o 

 in quibus maioris facilitatis caufla, diffcrentialc conftans du non 

 expreffimus, pofito autem vt fupra dx—pdu; dy=zp'(iu; 

 dd xzzzzdpduznqdu* et d dyzzzq' diC ', habebimus has aequa- 

 iioncs : 



q-\- ap + (3p' + y* + 5)' = c; ^ + a'/> + £'/>' + y'x + 5'j'no. 



Pro priori fupponamus multiplicatorcm effc (t>, pro poflcriori 

 autcm viV, tumque effe 



<P du (q-\- a.p+ (3/>'+y.Y+-5/) 4- \W« (tf '+ "'/>+ |3'/>'+ y'jf-4 ^Jy)— o, 

 quantitatcm integrabilem. Simpliciflima autem forma multi- 

 plicatorum (p et \1/ fine dubio crit ea, qua fupponuntur tantum 

 variabilem u continere, hac igitur fuppofitione facta, ob 



V= $ (q -f <xp + j3p' + y * -f 5?) + ^ (?'+ a'p+ £'/>'+ y '.*+ 5» 



• v = cp 



