a + b 



(fi - V \\ (6 cos a cos tj 



a - b 



+ a sin a sin 77)6 ^ - , 



2ff 



[84b] 



I a + b . . _^ r 



= - [/tI (6 cos a sin 77 - a sin a cos r;)e ^ + — (^ - ^^), [84c] 



or, by Equations [82x] and [83d, e], 



a + b Vt] 

 (^ = U — ; J (b cosa cos t] + a sina sin rj) - , 



a + b r a'-f b' 



ijj - - V — ; {b cos a sin tj - a sin a cos rj) + In 



a' + b' 



2n 



a + b 



[84d] 



[84e] 



These formulas hold at any instant provided the axes are drawn with the origin on the axis 

 of the cylinder and the ;c-axis along the major axis of its profile. The surface of the cylinder 

 is the ellipse ^- ^., or 0'= a, b' = b. 



The components of velocity, in directions given by Equation [82b', c'], are, from 

 Equations [82y,z], [84b] and [82x], 



a + b b cosa cos r} + a sin ct sin -q 



(6'^ + c^ sin^ 7;) 



2 ^\V2 



[84f] 



'it-^ 



{b'^ + c"- sin^ riY^ 



2 ^\y^ 



a+ b . r 

 U {b cos a sin rj - a sin a cos 77) + 



a'+ b' 



[84g] 



On the cylinder itself a'= a, b' - b. 

 On the a!-axis. 



and 



7/ = or 77, X ^ - a', 6'= yja'^ - c^ = yjx'^ - c^, 

 U b (a + b) cos a 



^-- % 



it 



^Ix^ - C^ {\x\ + yjx^ - C^) 



U a (a + b) sin a F 



On the y-axis. 



y/x^ - C^ (\x\ +y/x^ - C^) 2n -J X^ - C^ 



7? = 77/2 or 377/2, y = ± 6', a'= yf^~¥~c^ 



[84h] 



[84i] 



202 



