FORMVL. DIFFERENTULIVM. 149 



rcfiduum praebebit hanc aequationem : 



^R __(m+i)(w-2)^5 



±T 2 dx'^-'~^ I. 2 dx^-^' 



Deindc fi hinc Oibtrahatur 



3^R —^^^"^Zl^s^^ 



omerget noua haec aequatio : 



3^R 6'ddS m(m-i)d^'\J_ 



Q^''-r^'^'dT ""-~^'dx--^-^' 



Criteria igitur integrabilitatis formulae /^.v/</Ar/V</^, 



his tribus aequationibus continentur : 



JP ddO /R , ^""U 



dx d X d X — «^ 



2^(i 3ddR. 4^'S -~ md^^- ^V _ 



d X d X d X a X 



3 ^R <J.^JS I o /T ^lil ^'""'U ^ 



^ dx dx dx — 1, 2. ax 



•vbi euidens eft , in tertia harum aequationum coef- 

 ficientes numericos efle numeros trigonales. Si lam 

 formulae difFcrentiales proponerentur , quae plura 

 adhuc integralia inuoluerent , fine vlla dithcultate , 

 fingula earum crireria inte.:.rabilitatis aflignari pote- 

 runt , quemadmouum fi integrabilis efle debeat taUs 

 formula dxf^'^\dx, tum enim pofito 



T 3 ^V 



