168 



Tlie equations of observation are represented bv the straight lines 



Vi= aix 4 i^iy + ft,= (weight pi) {i = 1, ... «). 

 The point P(.r, y) is subjected to the force 



in which t), represents, in amount and direction, the distance of the 

 line T"/=0 to the point P. 



The point P remains at rest, if its coordinates satisfy the equations 



[/);«/ -J X + [pif(i^i]>l + [piCdHi] = 0, 



[Piii, oi] 'V + [pi^i'la ^ [pi^iin] = 0. 



Denoting here the potential U by z, we obtain 



Tiiis equation represents an elliptic paraboloid W, being the sum- 

 surface of tlie parabolic cylinders 



Pi{a{.v + i^iy 4- mY — 2?/, 

 wiiich have the phm c^O as sinnmit-(angent-plane ahmg tiie gene- 

 rator «, .r -f- ;?/// -|- fi/= 0, c r= 0, and which are obtained by trans- 

 kxting the parabola 



2 



,,.s -. 



/'/ 



lying in the normal plane of TV " «/-f -|- i/// + f'/= *\ perpendicularly 



1 

 to T/zzrO. The parameter of this parabola is — . 



Pi 



The summit T of the elliptic paraboloid V^ ([/), I",'] = 2c) is pro- 

 jected on c — into the point P, satisfying the normal equations. 



By constructing the tangent cylinder, the vertex of which lies 

 upon the .c-axis at intinity, we obtain a parabolic cylinder, the 

 perpendicular transverse section of which has a parameter equal to 

 the reciprocal value of the weight (/^ of the variable x. 



There being only two \ai-iables, only one (rigorous) equation of 

 condition (.r, y) = may be added; (,/•,?/) = represents (he 

 curve to which the point P is constrained. 



We now have, to determine that particular ellipse of the homothetic 

 set [/9/ Fi*] == const., which touches the curve 0. The point of 

 contact is the point P required. 



In 0, near the probable position of P, the new origin 0' is 

 taken. We have thus only to operate with linear functions of the 

 coordinates. So we really replace by its tangent E at P. 



Tlie elliptic paraboloid W is cut by the vertical of P in the 

 point »S. The vertical plane P,, which intersects * = along P, 

 pierces the paraboloid H' along the parabola V^,. having , Sas summit. 



We now construct the cylinder having its vertex at the point 



