OF THE MOTION OF A SYSTEM. 



fil5 



Lastly, while z,,, is substituted for its equal z,,y with 

 which it is identical, we shall have x^^, and y^^'p. the same 

 plane with x^^ and y^^, 

 which is that of the equa- 

 tor: we have thus 

 ^./=^...cos^ — y^^^sin^; 



[The second sign in the value of x,^ is here negative, 

 because the axis x^^ is not between ar^,, and y,,,, while y^, is 

 between y^^^ and x,,^,"] By substituting successively the 

 values thus obtained, we have [first 



x,-=^^.a cos <p—y,„ sin <p ; 



yz=.y^^^ COS ^ cos fl + x^^, sin ^ cos fl + s^^^ sin d; 



z,'=^z^^^ cos 5 — y^^^ cos (p sin 5 — x,^, sin ^ sin 5; then 



xzzx^i, cos ^ cos 4^ — y,j, sin ^ cos \^ + y,,^ cos ^ cos d sin -v^ + x^^, 



sin ^ cos d sin 4* + ^/// sin ^ sin ij/ ; 

 y=^y,„ cos ^ cos 6 cos 4' + x,„ sin ^ cos 6 cos ^ + z^^, sin d cos ^^ 



— x,^^ cos (p sin ^-hy,,, sin ^ sin 4^ ; 

 znz^^^ cos d— y,,, cos <f> sin 5 — o:^^^ sin (p sin 6; or, collecting 



the coefficients] 

 a:=x^,, (cos <p cos "4/ + sin <p cos sin -vl/) -f i/,^^ (cos (p cos 5 sin 4^ 



— sin (p cos 4') + «,^, sin 5 sin 4/ ; 

 y=x^,, (sin ^ cos 5 cos \J/— cos^ sin 4')-i-y//, (cos ^ cos 6 cos >|^ 



+ sin <p sin ij/) + z^„ sin 6 cos -^ ; 

 %■=. — x^,^ sin (p sin d — y^^^ cos ^ sin 5-fz,^, cos d. 



Corollary 1. We find also 

 x^^,:=.x (cos 5 sin •J' sin ^ -f- cos >J/ cos (p) 



■i-y (cos 6 cos \|/ sin ^ — sin -^ cos ^) — z sin d sin ^ ; 

 y,„'=.x (cos d sin 4/ cos ^— cos 4^ sin <p) 



-\-y (cos 6 cos 4/ cos ^ -j- sin 4^ sin (p)-^z sin 6 cos ^ ; 

 z^^^= X sin fi sin 4^ + y sin d cos '^^z cos fi. 



