5-2G MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



From (43) 



a 5 f a 3 / u Yl , 1 9V 

 sin gj [sin B ^(^- )\ + ao 3? = 5 



put 



then 



a 



sin 6 tan' - = u u , 



+ = cot 6 ~ + u (s* cosec 2 - cot 2 0), 



and 



sec" r > 



7 UVV . / 



- t fl _L * / t /) _i_ s 



do \ sin I 



to-n _ 



so that 



tan ^ 



cPu n s + cos 



9in s <T ' 



cPu n . du n f 2s cos l ' cos* (/ 1 . s* + s cos I s 



, u 



.dun_ f 2s cos Q 1 2 cos 2 0~] g 3 + s cos 



" d0 '- U S [f sin ^ ' " sin " siii" ^ J + "sin^^~ " 



hence, 



= oo r ff "1 



a 2 /c 2 = S (s 3 - s) A, tan'- cosec 2 0\ 



= 2 L * J 



Again, 



d f \ du n \ d fcos ^ + s 



~" 8 W 



_ (cos + s) 2 /2s cos 1_ 2 cos 2 <?\ ^ j 



sin 3 V sin 3 & sin siu* J sin 3 



hence, 



t = oo r ^ *icos^ 



a 2 Kj = S (s 8 s) A, tan* - cosec 2 6 , 



so that * 2 an( l *i cannot both vanish all along any curve drawn on the middle-surface, 

 unless the A vanish, which gives no displacement. 



We have shown explicitly in this particular case that the assumption that no line 

 on the middle-surface is altered in length does lead to expressions for the displace- 

 ments which cannot satisfy the boundary-conditions which hold at a free edge. 



