Vibrations of a Rod of Varying Cross-Section. 97 



Applying these to (11). we have 



and 





By differentiation o£ (7) and the use of (30) and (31) we 

 obtain 



A ^- B ^ =0 ( 32 > 



which on being combined with (30) to eliminate A and B 

 gives as the equation to determine # l3 and therefore \, 



dx\dx dx ) ^ ' 



By a process similar to that which was applied to (19) we 

 find that (33) is equivalent to 



= 



(« + 3) a (w + 4) l!(n + 3) 2 (n + 4) 2 (tt + 5)(>* + 6) 



+ 



x 4 



2!(n + 3) 2 (72 + 4) 2 (7?.+ 5) 2 (n-t-6)(n + 7)(w + 8) 

 which if n is integral may be written 

 1 x 2 



= 



(w + 3) !(n + 4) ! " 1 !(n+4) !(n + 6) ! 



,r>' 



+ 2! (n+5) !(»+8)! ^ 



7. In the case of a wedge-shaped rod we have ?i = 0. The 

 equation to determine x 1 is 



1 x 2 x* x 6 



3! 4! 1!4! 6! ' 2! 5!*! 3 ! 6 ! 10 ! ""' 

 the lower roots of which are found to be 



12*757, 27-755, 47-576, 72 294. 



AI — {©X*W+($).W«)} • • • (35) 

 _PAtZ. 3%. S. 6. Yol. 25. No. 145. Jan. 1913, H 



