889 



Substituting these expressions in the formula (30), we get the results 

 collected in the next table. 



6. With regard to future developments we shall determine the 

 derivatives of the functions il^ and ^, (considered as functions of 

 p,, <j,, q and t) with respect \o q; to this purpose we need the 

 value of the function 



«/ 



1 . 



bq \Jbe 

 which value is given by the equation 



Ö 



be 



iPo^ffla L 





J/OO'O^O 



a^ 

 d^ 



(37) 



For the determination of the function 



b eoa2 to 

 q — :;-;; — = — 4 «in u», 



dedO 



we have the formulas : 



dó 



d cos A w 



do 



d cos 6 w 



do 



Hence 



= -|- 8 sin w — 8 sin 3 ly, 



= — 12 sin w -\- 12 sin '6 w — 1 2 n'n 5 w. 



1 0* • 9 T-T- = + 2092 sin rv 



(38) 



(39) 



— 156 sin 3 w 

 — 12 sin 5 w. 



For the development of the goniometric functions of the different 

 multiples of iv as functions of t, in the first place we have the 

 formula (Investigations p. 27) : 



tin to = 4 0.99841 sin r — 0.00159 sin 3 t . 



