| T = ½p (x12 + x22) + ½p′x32 + q (x1y1 + x2y2) + q′x3y3 + ½r (y12 + y22) + ½r′y32 |
(1)
so that a fourth integral is given by
dy3 / dt = 0, y3 = constant;
(2)
| dx3 | = x1 (qx2 + ry2) − x2 (qx1 + ry1) = r (x1y2 − x2y1), |
| dt |
(3)
| 1 | ( | dx3 | ) | 2 | = (x12 + x22) (y12 + y22) − (x1y1 + x2y2)2 |
| r2 | dt |
| = (x12 + x22) (y12 + y22) − (FG − x3y3)2 = (x12 + x22) (y12 + y22 + y32 − G2) − (Gx3 − Fy3)2, |
(4)