218 L EULERI OPERA POSTHUMA. Asiron.mech, 



ad rectam Gxam /fE normali YXy sint ternae coordinatae AX=x, XF=j, YB = z\ ipsa autem] 



distantia ^5 = c = V (ccaj -i- j^ -+- zz). Cum jam vis, qua B ad J urgetur sit = ^ > massas 



corporum per litteras A eX B indicando, ea secundum directiones ternarura coordinatarum resolutal 

 dabit vim in directione 



.^ —B{A-t-B)x 

 A\ = = f 



yy_ -B(A-i-B)y 

 Y^ __ -B(A-t-B)z ^ 



unde sequentes aequationes elicimus sumendo elementum dt constans 



^ .£ .IIiv 



ddx = -^'^^^^^^^ ^dt\ ddy = -^'^^^^^^' >dt\ ddz = -^'^^,-^^^'.dt\ 



Hinc dt^ eliminando colligimus 



yddx — xddy = , zddy — yddz = , xddz — zddx = , 



quarum quidem quaelibet in binis reliquis jam continetur, ita ut duas tractasse sudQciat. Inde ergl 

 integrando obtinemus 



ydx — xdy = Edt et zdy — ydz = Fdt , hincque 



•'..'•? ',.. . -> h,]ia P{y^^ — ^dy) -»- E {ydz — zdy) = 0, 

 quae per yy divisa et integrata dat 



1 1- G = 0, seu Ez -t- Fx -i- Gy = 0, 



m 



e\ qua liquet motum corporis B fieri in plano per punctum A transeunte. Cum igitur habeamus 

 Ez -\- Fx -\- Gy = () y ac praeterea has tres aequationes differentiales 



Edl = ydx — xdy , Fdt = zdy — ydz , Gdt = xdz — zdx , 

 .0/» ih 

 ob xx-i-yy-^zz = vv adipiscemur quadratis addendis 



i 



(EE-i-FF-^ GG^dt^^dx^^vv — xx^-^dy"^ (w — yy) n-dz^ {vv — zz) — 2xydxdy — 2yz dydz — 2£cz dxdz, 



et quia xdx -*- ydy -i- zdz = vdv, obtinebimus 



{EE-i- FF-^ GG) dt^= vv {dx"-^ dy''-i-dz'') — vvdv''. 



Verum si aequationum differentio-differentialium prima per 2dxy secunda per 2dy et tertia per 2dz 

 multiplicetur, summa erit 



2dxddx -h- 2dyddy -h 2dzddz=^^^^^^^^^'dt\ 

 cujus integrale, oh d( constans, est 



dx^-i.dy^-^dz^^^Ddt^^-^^^^^^dt^ 

 qui valor ob superiorem aequationem est 



