#4: Mr. E. R. Darnley on the Iransverse Vibrations 

 Then c r + d r = a r $> r — a r+1 ^fr r , (6) 



and, similarly, if sin KZ r+ i is not zero, 



By (4) 



(7) 



cos K/,. + rf r cosh l\l r = a r+ i cot K/ r — <z. 



— a r+1 eothKZ r + a, 



cosh 2 KZ r 



cos 2 KZ, 

 sin K/ r 



sinh KZ,. * 

 and by (2) and (7) 



a r+ i<j> r+1 - Or+^r+i + «r(sin KZ r 4- sinh KZ,.) 



T . 7 7 , T ^ 7 , cos 2 KZ r cosh 2 KZ,. 



= (; r cosji.6 P + a r cosnKA P = — fl r+1 9 r — a? — •" "^77 "^ a ^ 



wh 



ence 



or 



sin K/ r r sinh Ki r r 



, , , f / . Tr7 cos 2 K/A 



a r+ i(</>, + ^r+i)— ar+a^r+i + «r j ^sin KZ,+ ginK/ J 



+ ( shlh K/,-^^'-)}=0, 

 V sinh KZ ? . J ) ^ j 



§ 5. If the end of the first bay be free, the equation for 

 this bay will be 



u=a 1 (cos K«i'+ cosh K#) + e^sin Kx-\- sinh Ka?), 



, . . rf 2 « , </ 3 « 



since at the origin - ■ ■— and -=-= are zero. 

 ° rfar dor 



The conditions at the junction of the first two bays are 

 a^cosK/i-f cosh K/j) +e 1 (sin KZ X + sinhKZ^^O, 



«i( — sin KZj + sinh K/ x ) + c t (cos KZ X f cosh KZ X ) = c 2 4- 6? 2 ,, 



aj (cos KZ X — cosh KZ X ) + ^(sin K/ 2 — sinh KZ X ) = 2a 2 , 



whence 



7NJ f 2(1+ cosh KZ, cos K/Ol 



v yr { sinh I\Z 1 sin k/x J A - 



But e 2 + d. 2 = a 2 $>. 2 — a-^ 2 . 

 Hence 



Os^+^s)— ff8^« = °- 



§ 6. Equations giving the periods or whirling speeds can 

 dow be obtained as determinants. These determinants are 



