where X3 is a quartic function of x3, and thus t is given by an elliptic integral of the first kind; and by inversion x3 is in elliptic function of the time t. Now
(x1 − x2i) (y1 + y2i) = x1y1 + x2y2 + i (x1y2 − x2y1) = FG − xy3y3 + i √ X3,
(9)
| y1 + y2i | = | FG − x3y3 + i √ X3 | , |
| x1 + x2i | x12 + x22 |
(10)
| d | (x1 + x2i) = −i [ (q′ − q) x3 + r′y3 ] + irx3 (y1 + y2i), |
| dt |
(11)
| d | log (x1 + x2i) = −(q′ − q) x3 − r′y3 + rx3 | FG − x3y3 + i √ X3 | , |
| dti | F2 − x32 |
(12)
| d | log √ | x1 + x2i | = −(q′ − q) x3 − (r′ − r) y3 − Fr | Fy3 − Gx3 | , |
| dti | x1 − x2i | F2 − x32 |