| (55) | dx | = - | C q sec i | , |
| dq | f(q sec i) |
and multiplying by dy/dx or tan i,
| (56) | dy | = - | C q sec i tan i | ; |
| dq | f(q sec i) |
also
| (57) | di | = | Cg | , |
| dq | q sec i . f(q sec i) |
| (58) | d tan i | = | C g sec i | , |
| dq | q . f(q sec i) |
from which the values of t, x, y, i, and tan i are given by integration with respect to q, when sec i is given as a function of q by means of (51).
Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v2/k or v3/k.
But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their mean values, η, cos η, and sec η, especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.
Replacing then the angle i on the right-hand side of equations (54) - (56) by some mean value η, we introduce Siacci's pseudo-velocity u defined by