.UK MI;. L. x. (;. FILOX ox Tin: KKSISTANCK TO 



Having obtained ', the shears are easily deduced by simple differentiation 



xz 1 / . . .. . dw . ^ Aw\ . , ,. 



= (sinh t sin 77 . + cosh f cos 77 , TC sinh f cos 77 



y. e-l \ if i; (it)/ 



yi. 1 / ,. dw r > <hc\ i f 



= - ( cosh f cos 77 -7.. sinh f sin 77 --- -- -4- re cosh f sin 77 



/t rJ \ ' fl% ' <^/ 



where J = cosh 2 f cos 2 77 -f sinh 2 sin 2 77. 



These again may be put into the slightly different form 



** 1 /.,,.. rfw, . , ,,. rfw,\ i . /, 



- =r y (sinh f sin 77 --- -f- cosh f cos 77 -j-\ TC. sinh f cos 77 (1 -f- sech 



yz If , .. <ln\ \ f dw\ ^ * /, 



- = -=-j cosh f cos 77 sinh f sin 77 I + TC cosh f sin 77 (I sech 



The next quantity which we require is the moment of the shears 



r ^-~ ^ 

 M = \(xyz yxz) dx dtj 



- f c?77 f (/^[(cosh 2^ cos 277) sech 2a (1 cosh 2^ cos 2ij)] X J 



Jfl' J - a 



UTC 1 [P C a 



= \ dr)\ dg (cosh 4 cos 4i; sech 2a (cosh 2f + cos 277} 



" J/3' J -o 



+ sech 2a {cosh 4^ cos 2?j + cosh 2f cos 4ii}) 

 + ^ f%^ sin 27, [V, J - ^ J" d sinh 2^ [IP,]^ 



y sinh 4a a sin 4y cos 4e sech 2a (2-y sinh 2a + 2a sin 2y cos 2e) 



== ~5~ sech 2 . . , 



H ^ (sinh 4a cos 2c sin 2y + sinh 2a cos 4e sin 4y) 



IG^rc'a* (cosh 2 + cos 2 cos 2 7 ) f ^ . . 2 + lir / 



Sinh ^~ V) ~ e ) sin 2> ? *9 



7r(2 + l)[w(2w + l) s + 16a 2 ]cosh f!LjiE? ' 



at 



ft 2w -4- ITT 



cosh >j x (77 e) sin 277 ^77 



16] sinh 



2a 



TV (cosh 2a + cos 2e cos 2?) ( - 1 ) tanli 2 " 



sinh 



