and also
Further [we obtain], from the result obtained by differentiating (8),
But we can derive two other expressions for this. We have
therefore [because of (8)]
[and therefore, from (7),]
After these preliminaries [using (2) and (4)] we shall now first put
in the form
and differentiating with respect to
we have[1]
![{\displaystyle {\begin{aligned}{\frac {\partial m}{\partial s}}&={\frac {\partial x}{\partial \theta }}.{\frac {\partial \xi '}{\partial s}}+{\frac {\partial y}{\partial \theta }}.{\frac {\partial \eta '}{\partial s}}+{\frac {\partial z}{\partial \theta }}.{\frac {\partial \zeta '}{\partial s}}+\xi '\,{\frac {\partial ^{2}x}{\partial s\,\partial \theta }}+\eta '\,{\frac {\partial ^{2}y}{\partial s\,\partial \theta }}+\zeta '\,{\frac {\partial ^{2}z}{\partial s\,\partial \theta }}\\&=mp'(\xi 'X+\eta 'Y+\zeta 'Z)+\xi '\,{\frac {\partial \xi }{\partial \theta }}+\eta '\,{\frac {\partial \eta }{\partial \theta }}+\zeta '\,{\frac {\partial \zeta }{\partial \theta }}\\&=\xi '\,{\frac {\partial \xi }{\partial \theta }}+\eta '\,{\frac {\partial \eta }{\partial \theta }}+\zeta '\,{\frac {\partial \zeta }{\partial \theta }}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3fa689fa228d062e9cbf2503c3ba9ac1485bc9a)
- ↑ It is better to differentiate
[In fact from (2) and (4)
therefore