AND DEFORMATION OF A THIN KLASTIC SHKI.l. 529 



Then, for u, v, w as functions of 6 it is convenient to take equations (51), (54), and 

 (56), which become 



du , . .,-, 2cos0 . if dw . _. 



]u--- V +-- = 0, . (57) 



+ cot0^ + [2+K* {I +s 2 (l + c)}cosec ! 0]r 



}, (58) 



dff 3 



. scos 6 sc du 



Bin 



(59) 



Differentiating (59) witli respect to 6, dividing by c and subtracting from (57), 







/o .2 rcos^ * dv 



(2 + ^ .- 



Write u sin ^ = U, v sin = V, thus, 



f [(2 + <> sin - ,'] ^, = . - . + 9 in ;.. . (60) 



and (59) becomes 



rfU V 



We are going to substitute from (60) in (58) and (61) ; the result will enable us to 

 eliminate V, and obtain an equation for w. 

 We have 



* /, 1\ a dw , O d\ 



- 1 1 + - 1 sm* -- + s . sin - 

 TT 2 \ c) <16 d0 . 



(2 + ) sin 8 - * 



therefore 



rfU / 1\ sin'g f A**, a dw 2 (2 -f ) sin* g cos g 



= ~ 1 h - 811 + "--* 



pPV fl rfV 2(2 



" 1 " 



Substituting in (61), we have, on multiplying by cosec 0[(2 -f- *c j )sin 8 6 a 2 ] 8 , 



MDOOCLXXXVIII. A. 3 Y 



