§. 13- Si igitur propofita fuerit haec aequatio dif- 

 ferentialis : 



d X d y 



a-t-2bx-i-cxx a -{-2 by -t-cjy ' 



cius integrale conipletum ita euit exf^reffiim : 



(_a-l^ ' b x-^-cx x) (a-t-tby-i-cyy) r^ 



quae, vtrinque addendo b b — a c, induet hanc formam: 



aa-\-zab{x~i-y]-i-7.acxy-\-bbix- i -y)'^-^ibcxy{x-^- y ) -(- - c x xyy _ 



(X — >"p -^A, 



ficque, extrada radice, integrale hanc formam habebit: 



a -^bf,x -i-y)-{-cxy a 



lE—y ^ > 



quae fme dubio eft fimpHcifrima, qnandoquidem tam y 

 per X quam x per y facilhme exprimi poteft , cum fit 



V [ A — b)x—a -^. g-4-( A-t-&)3' 



^ A-^-b-^cx ^" '* ^_t, — cy ' 



§. 14. Calculum , quo hic vfi fumus, perpendenti 

 facile patebit, in his formis X et Y , non vitra quadrata 

 progredi licere. Si enim ipfi X infuper tribuamus termi- 

 num d A"' et ipfi Y termijmm dy , pro priore forma piodit 



f^^-^b + c{x ^^j) -hd[x X + xj ^yy)-^; 

 pro altera autem forma efl 



X'-^Y'-^b+zc{x-Vy)^^d[xx-\-yy)-'^^-\-i^^, 

 Quare fi hinc duplum praecedentis aufferamns, colUgitur 



^i^ _i- ^JJL. — l±L — d ( x—vY' 



quam aequationem non amplius inregrare licet» 



§. 15. Facilc autem oflendi potefl , talem aequa- 

 tionem differentialem, in qua vJtrd quadratum proceditur, 

 nullo amplius modo algebraicc integrari pofle. Si enim 



tan- 



