104 CELESTIAL MECHANICS. I. i. 3. 



retain the variations ^r ?nd ^/, and to s'lbsti'u^e for them 

 their values derived from the nature of the surface, since 

 we are thus en bled to determine the pressure. 



Scholium 2. Now, if a, 6, and c be the coordinates 

 of the origin of the perpendicular r, for the part of the sur- 

 face ill question, without any regard to this origin remain- 

 ing as a fixed point, we have tlie equation r^-=z{x — a)^ -f 

 {ij — hy +(z — cY, supposing only that a, &, and c remain 

 constant for an elementary portion of the surface, as they 

 must do in all cases : we have, then, for Zx^ 2rSV=i2(x — a) 



^ J ^'r x — a 1 .,, ^V_v— & J 



ex, and ^^= , and m the same manner — =^- ,and 



hx r dy r 



____; and since (_-) +{t-) +(_.)^=l.we 



have consequently (k— )^ "^(^^)^"''(f ) ^— ^' 



[Scholium 3. The substance of these scholia is ex- 

 pressed by the author in a form somewhat different; and in 

 order that no injustice may be done to the symmetry of his 

 system, it will be proper to insert his reasoning in its ori- 

 ginal form, with some explanatory remarks. " Let w=:0 

 be the equation of the surface, then the two equations Sr 

 =0 and Sm=0 will both be true together, which implies 

 that 8r may be ■=lNZu, iV being a function of Xy y, and z. 

 In order to determine this function, the coordinates of the 

 origin of r may be called a, b, and c, we shall then have 



r=:>/ < (a: — a)^ +(y—by +(z~cY >, whence we obtain 

 + (^Y-^(^Y]-1; so that if we make >.zzR : ^ 



