Vibrations : Both Masses and Periods Unequal. 

 Further, the coupling may be written 



- = £e 



1 (i + p +/3p)(/3 + v -i- V p)' ' ' 

 To simplify, (9) and (10) may be abbreviated thus : 



3 +*£+*-* 



d 2 z 



d? 



+ 



Where 



/3 + i] + T)p + /3r)p 



and c 



Sm 2 



fi + n + Tjp 



39 



(ID 



(12) 

 (13) 



(14) 



(15) 



Equations (12) and (13) are the same as (6) and (7) of 

 Paper II., but the values of a, b, and c are different. 



Solution and Frequencies. — To solve (12) and (13) we 

 write 



z — e 3 *- \ 



Then ( 



From equation (11) of Paper II., we see that the values 

 o£ x may be written 



— r + ip and — s + iq (16) 



Hence, omitting small quantities, we have 



2 q 2 =[c + a-i/{(a-c) 2 + 4pb 2 \]\ 



_ a-c+v{(a-6') 2 -+4^ 2 } 7 -j 

 r ~ 2j{(a-cf+4,pb 2 } 



n _ c-a4- x /{(a- c ) 2 + ±pb 2 \ 



(17) 



(18) 



2i/U«-c) 2 + 4^ 2 } 



Thus using (15) and (16) and introducing the usual con- 

 stants, the general solution may be written in the form 



z==e- rt (Ae pit + Be- pit ) + e- s XCe qit + ~De- qit ), . (19) 

 and 



y= C-^ + c ) g -^ ( A^ + B g -^) + ti 



st (Ce qit +De- qit ) 



2pri 



e-^-A^ + Be-**) + ?8^-*((V* + D«- 



gr«, 



(20) 



