AND DEFORMATION OF A THIN ELASTIC SHELL. 539 



Let u a cos s<f>, v a sin s<j> ; then fur w and r as functions of z we have the equations 



Let u^, u 2 * be the roots of the quadratic (95), then 



u = cos *< e'f' [ P l cos u t z + P 2 cos /ijZ P^ sin 

 v = sin s<> e'^Q sin uz + Q sin uz + Q' cos 



P' 2 sin 

 + Q' 2 cos 



so that 



= cos s 

 + cos .s 



sn 

 j) cos 

 3 z 2 



sn 

 cos) 



- _ 



_ + I), - C - = 0, 

 where 



A = 



rf 



........ (93) 



c- '-i 4. 2 * l + 



-a( } r l-<r 



b.k-#i*- 



f \ /8 l-<r/ J 



To satisfy these, take 



t* = P cos 02 ] if = f sin u2 ] 



J- or } ...... (94) 



v=Q8in/i2j v = Q cos uz I 



Then 



P (B - A/i 2 ) + QC/t = 0, 

 ( ,. . PCu-f 



F 



whence 



(B - Au) (D - u) - C 3 ^ = 0. 



This is a quadratic in u 8 , viz. : 



A /i *-(B + AD + C 2 ) /A 2 + BD = 0; ..... (95) 

 and we have 



Q = C - P '' Q' = C.P'.. ..... (96) 



. (97) 



. (98) 



