\begin{array} { r l r } { C _ { 3 0 } ^ { \prime } } & { = } & { { \textstyle \frac { 1 } { 8 } } \Big ( 3 \cos \theta + 5 \cos 3 \theta \Big ) C _ { 3 0 } - { \textstyle \frac { 3 } { 4 } } \Big ( 3 + 5 \cos 2 \theta \Big ) \sin \theta \Big ( S _ { 3 1 } \cos \psi + C _ { 3 1 } \sin \psi \Big ) - } \\ & { } & { - \, 1 5 \cos \theta \sin ^ { 2 } \theta \Big ( C _ { 3 2 } \cos 2 \psi - S _ { 3 2 } \sin 2 \psi \Big ) + 1 5 \sin ^ { 3 } \theta \Big ( S _ { 3 3 } \cos 3 \psi + C _ { 3 3 } \sin 3 \psi \Big ) , } \\ { C _ { 3 1 } ^ { \prime } } & { = } & { { \textstyle \frac { 1 } { 8 } } \Big ( 3 + 5 \cos 2 \theta \Big ) \Big ( C _ { 3 1 } \cos \psi - S _ { 3 1 } \sin \psi \Big ) - { \textstyle \frac { 5 } { 2 } } \sin 2 \theta \Big ( S _ { 3 2 } \cos 2 \psi + C _ { 3 2 } \sin 2 \psi \Big ) - } \\ & { } & { - \, { \textstyle \frac { 1 5 } { 2 } } \sin ^ { 2 } \theta \Big ( C _ { 3 3 } \cos 3 \psi - S _ { 3 3 } \sin 3 \psi \Big ) , } \\ { C _ { 3 2 } ^ { \prime } } & { = } & { - { \textstyle \frac { 1 } { 4 } } \cos \theta \sin ^ { 2 } \theta \, C _ { 3 0 } - { \textstyle \frac { 1 } { 8 } } \Big ( 1 + 3 \cos 2 \theta \Big ) \sin \theta \Big ( S _ { 3 1 } \cos \psi + C _ { 3 1 } \sin \psi \Big ) + } \\ & { } & { + \, { \textstyle \frac { 1 } { 8 } } \Big ( 5 \cos \theta + 3 \cos 3 \theta \Big ) \Big ( C _ { 3 2 } \cos 2 \psi - S _ { 3 2 } \sin 2 \psi \Big ) - { \textstyle \frac { 3 } { 8 } } \Big ( 5 \sin \theta + \sin 3 \theta \Big ) \Big ( C _ { 3 3 } \sin 3 \psi + S _ { 3 3 } \cos 3 \psi \Big ) , } \\ { C _ { 3 3 } ^ { \prime } } & { = } & { - { \textstyle \frac { 1 } { 8 } } \sin ^ { 2 } \theta \Big ( C _ { 3 1 } \cos \psi - S _ { 3 1 } \sin \psi \Big ) - { \textstyle \frac { 1 } { 4 } } \sin 2 \theta \Big ( C _ { 3 2 } \sin 2 \psi + S _ { 3 2 } \cos 2 \psi \Big ) + } \\ & { } & { + \, { \textstyle \frac { 1 } { 8 } } \Big ( 5 + 3 \cos 2 \theta \Big ) \Big ( C _ { 3 3 } \cos 3 \psi - S _ { 3 3 } \sin 3 \psi \Big ) , } \\ { S _ { 3 1 } ^ { \prime } } & { = } & { { \textstyle \frac { 1 } { 8 } } \sin \theta \Big ( 3 + 5 \cos 2 \theta \Big ) C _ { 3 0 } + { \textstyle \frac { 1 } { 8 } } \cos \theta \Big ( 1 5 \cos 2 \theta - 7 \Big ) \Big ( C _ { 3 1 } \sin \psi + S _ { 3 1 } \cos \psi \Big ) - } \\ & { } & { - \, { \textstyle \frac { 5 } { 8 } } \Big ( \sin \theta - 3 \sin 3 \theta \Big ) \Big ( C _ { 3 2 } \cos 2 \psi - S _ { 3 2 } \sin 2 \psi \Big ) - { \textstyle \frac { 1 5 } { 2 } } \cos \theta \sin ^ { 2 } \theta \Big ( S _ { 3 3 } \cos 3 \psi + C _ { 3 3 } \sin 3 \psi \Big ) , } \\ { S _ { 3 2 } ^ { \prime } } & { = } & { \cos 2 \theta \Big ( C _ { 3 2 } \sin 2 \psi + S _ { 3 2 } \cos 2 \psi \Big ) + { \textstyle \frac { 1 } { 4 } } \sin 2 \theta \Big ( C _ { 3 1 } \cos \psi - S _ { 3 1 } \sin \psi \Big ) + { \textstyle \frac { 3 } { 2 } } \sin 2 \theta \Big ( C _ { 3 3 } \cos 3 \psi - S _ { 3 3 } \sin 3 \psi \Big ) , } \\ { S _ { 3 3 } ^ { \prime } } & { = } & { - { \textstyle \frac { 1 } { 2 4 } } \sin ^ { 3 } \theta \, C _ { 3 0 } - { \textstyle \frac { 1 } { 8 } } \cos \theta \sin ^ { 2 } \theta \Big ( C _ { 3 1 } \sin \psi + S _ { 3 1 } \cos \psi \Big ) + { \textstyle \frac { 1 } { 1 6 } } \Big ( 5 \sin \theta + \sin 3 \theta \Big ) \Big ( C _ { 3 2 } \cos 2 \psi - S _ { 3 2 } \sin 2 \psi \Big ) + } \\ & { } & { + \, { \textstyle \frac { 1 } { 1 6 } } \Big ( 1 5 \cos \theta + \cos 3 \theta \Big ) \Big ( C _ { 3 3 } \sin 3 \psi + S _ { 3 3 } \cos 3 \psi \Big ) . } \end{array}