498 Mr. H. Holt on a Method of Calculating 



(5) Value of ^m l -^cos2(v — v J ) : 

 Pi 



Pi 



2(v— ^) = c^ -h 2D Z sin c t t -, 



.'. cos 2(v — vj) = cos[C]/4- XDi sin Cit] 

 (by Taylors theorem) 



= cos c x t— sin cf . 2)D ? sin c^ 



= cosc^ + 2iDiCOs (Cj + c z )£ — £-^D z cos (cj — cjtf ; 



.*. %m i -^gcos 2(v — ^) = §& cosc^ + S| £(B Z + F z + D z ) cos (c x + c } )t 

 Pi 



+ 2§A(B z + Fi-Dj)cos(c 1 -c z )f. 

 Hence, collecting these values, equation (7) becomes 



0= Z\_(c? + P + 3)F Z + 2c z G z ] cos ci/ + S\ kB t cos cjf, 

 + %kcos c x t 



+ ^t*^+ft + D l )cos(c 1 + .c,)/ 

 + ;£f £(B Z + F Z -D Z ) cos fa-Cij* 

 + ^feG^-f RF^ + i^iG^GL) cos (cj + c m )f 

 Applying this equation to the case of any given argument cj, 

 we shall have 



-}--§& in the case when ^7=1, 



+ |£(Bj + F z + D z ) when c x = Cl + Ci. 



+ f£(B z + F z -D z ) when c^Cj ~c b 



+ 2(c z G z F m - |FiF m + ic z G z c w ) when c,=C|±c m . . . (A) 



9. Again, substitute the assumed values of p and v in the dif- 

 ferential equation (8). 



(I) Value of ^2 P ^} 



p 2 = 1 +22F,cos dt + ^Wm cos [ct±cj)t, 



2 C ^=2 + Z2c l G l cosc l t; 

 at 



