454 L. EULEKl OFKUA FOSTHUMA. Anaiym. 



xdxVp ydyV p ,y 



j3 -+- yj/ H- §35 p-t-yx-*-8y ' 



seu- xdx{/3'^yx-*-8y)-^ydy{/3-+-yy-\-dx) = ^{i3-^yy-h-Sx){^-i-yx-^dy): at est 

 {^-i-yy-t-8x){^-^yx-^dy) = /3/3-^/3{y-\-d){x-^y)-i-yd{xx-\-yy)-t-{yy-^88)xy. 



vn. 



Quo hanc formulam facilius expediamus, ponamus x-^y = t et xy = u, erit 

 xx-i-yy = tt — 2u et x^-^y^=t^ — 3fa, 

 sicque aequatio abit in hanc formam 



/3{xdx-^ydy)-^y{xxdx-^yydy)-i-Sxy{dx-^dy)=y-{^/3-^/3{y-*-d)t-^ydtt-¥-{y^8)^u). 



Ipsa autem aequatio assumta fit: = a-h-2/3t-i- ytt-^-^^d—y^u, et penitus introductis litteris 

 « et « habebimus 



/3{ldt^du) -\- y {ttdt^tdu^udt) -+- dudt = ^{/3/3 — cc8-\- ^{y—8)t-i- {yy^bb)u), . 



seu dt{^t -V- ytt — {y—8)u) — du{^-^yt) = ^- {/3^ — ad -i- /3 {y-^d) t -^ [yy—dd^u). 



VIIL 



f^,i^ Ex aequatione autem assumta si differentietur, fit dt{/3-¥-yt) = {y — 8)du, unde aequationis 

 ultimae prius membrum transformatur in 



' ^^{-/3^-^^{y-^8)t — y8tt-{y—8)'u): 

 quod cum aequale esse debeat huic formulae 



. ' . ^{^^-i-^{y-^8)t-^y8tt-^{y—8)^u), 



commode inde oritur 5,,.; y- = ^^ et r= ^^^- 



IX. 



Cum jam sit t = x-t-y, habebimus sequentem aequationem integratam 



r ocdx r ydy^ p (x -t- y) Vp 



Jv{A-*-^Bx-t-Cxx) JV(A-H<iBy-*-Cyy)~ ^ 7-S ' 



existente = u-i-2^ x-i-y) -^y{xx-^yy) -¥-28xy, siquidem relatioues supra exhibitae inter 

 litteras ^, B, C et a, /3, y, 8 ^c p locum habeant. Hinc ergo eadem manente determinatione 

 variabilium cc et y erit generalius: 



r dx('!H- +- l8x) r dr/(^-4-25y) ^ ^(x-*-y)Vp 



JV(A -+- *iBx^Cxx) JV(A-i-<iBy-¥- Cyy) * y-S ' 



X. • 



Progrediamur porro, ac statuamus 



xxdx yydy ^y 



V^A-^^-lBx-^-Cxx) V(A-^'-lBy-^Cyy)~~ ' 



