OF THE MOTIONS OF A SOLID BODY. 247 



d%^. COS ^fi + d^ COS 6; which is the coefficient of C in the 

 value o£ x'dy — y'dx\ 



We have next to perform a similar computation for the 

 areas x'dz — zfdx', and y'dz' — z'dy' : and the same charac- 

 ters may again be employed in each of these cases with 

 their appropriate significations : half of them retaining the 

 same values. 



a zn cos 9 sin ^ sin <p -\- cos ^ cos q> 

 a 1Z — d^. sin 9 sin x|/ sin <p 4- d^^ (cos 6 cos 4' sin o— sin r|^ 



cos (p) 

 -f d^. (cos 9 sin 4^ cos ^— cos 4' sin (p) 

 ^zz — sin fl sin <p 



yzz —do, cos 9 sin <p — d<p. sin 9 cos <p 

 ^ =: cos 9 sin -^ cos ^— cos 4' sin <p 

 ^=: — dd. sin 9 sin 4^ cos ^ + d4' (cos 9 cos 4' cos ^ + sin 4^ 



sin <p) 



— d^ . (cos 9 sin 4^ sin ^ + cos 4' cos (p) 

 8 zr — sin 6 cos (p 



zz=. — d9. cos 9 cos ^ + dip. sin fl sin ^ 

 y = sin ^ sin 4' 



y'rz d9. cos 9 sin4' + d4'. sin 9 cos 4' 

 C = cos 9 

 ^= - d5. sin 9 

 aVi=L — d6.(cos25 sin ^ sin ^^-fcos 6 cos 4^ sin cos 9). 



— d^. (sin cos 9 sin 4^ sin cos ^ + sin 6 cos 4^ cos ^(p) 

 a^zz d9, sin H sin 4^ sin ^<p — d4'. (sin cos 9 cos 4^ sin V— 



sin 9 sin 4^ sin cos (p) — d^ (sin cos 9 sin 4^ sin cos cp — 



sin 9 cos 4^ sin ^2^) 

 a^-^a'S^: — do. (sin 4^ sin *<p + cos 5 €0S 4^ sin cos ^) 



-f d4' (sin cos 9 cos 4^ sin ^<p — sin 5 sin 4^ L _ ., 



sin cos <p) """ 



— d^. sind cos 4^ 



