TRANSFORMATION OF SURFACES BY BENDING. 105 



Also 08 = . ., 

 sin Sop 



1 



= aS(Sop) -,- 



Substituting the values of a, b, c, d from Art. (11), 

 81= - 



, 1 ds ,~ v 1 ds' 1 , 



= I -y- COt <}>OU I -- j ; - r OU , 



r du r du sin <p 



Finally, substituting the values of I, I', and 8Z from Art. (14), 

 d I 1 cfe'\ ,, 5. , cot $ ds' \ ds ~ ~. , 1 efe 1 efe' 



5 , 3-7) OU OM = -- r-Z-i -j, -j-bubu -- , . -5 -- -y , 



du \p sin (p du / p sin <f> du r du p sin $ du r du 



which may be put under the more convenient form 



d n * d , /I cfo'\ , 1 cfo . p 1 ds 1 



j- (log p) = -j- log -r f -j-, H -- 7 T- COt <A + -S - j -- : - 7 J 



du ^ du 3 \sin $ du) r du p r du sin < 



and from the value of SZ' we may similarly obtain 



d ,. d , /I <M , 1 ds' , a* 1 ds' 1 



-T~, (lOff P ) = I"/ lff - 7 J- + - J~? COt * + -j-, -. - 7 . 



du ' dtt ' \sm ^ dw/ r du p r du sin (j> 



We may simplify these equations by putting p for the specific curvature of 

 the surface, and q for the ratio -^ , which is the only quantity altered by bending. 

 We have then 



PP 



whence p* = < 

 and the equations become 



-j- (log a) = -T- log ( p -j- ) + , -j- cot <4 H -v 7 q, 

 du v 5 - 1 ' rftt ' \* c?w 1 1 r du r du sin 9 



rf , / ~ds^\ 2 dsT 2 ds' 1 1 



In this way we may reduce the problem of bending a surface to the 

 consideration of one variable q, by means of the lines of bending. 



14 



VOL. I. 



