Mr. W. H. L. Russell on the Calculus of Symbols. 431 

 and suppose that 



= A,L,,_2 + ^oM,„_2 + v,]Sr„_2 = &C. 



= &C. 



Then we find 



+ G,„H,„_iG'.,_.o+ H,.H'_,G',„_2+ K,„H",,_2G',,_2 

 + G"„iK,„_iG"„i_o+ H„,K'„,_iG",„_oH- K,„K"„j_iG"„i 



^3= GrmG-,„_iG„j_2G„j_3 + H,,,G'„,_iG„,_2G,„_3 



+ K„,G",„_iG,„_2G,„_3 + G,,,H„.,_iG'„,_2G„,_3 

 + H,„H'_,G'_2G,„_3 + K, jr'_2G'„G_3 

 + Gr»«K,„_iG",„_2G,„_g4-H„,K',„_iG"„,_2G,„_3 

 + K,„K",„_iG",„_2G„i_3 + G,„G,„_iH,„_2G',„_3 

 + H,„G',„_iH,„_2G',„_3 + K,„G",„_iH„,G'„,_3 

 + G,„H_iH'_2G'_3+ H,„H'_,H',„_2G',„_3 

 + K,„H",„_iH',„_2G',;,_3 4- K„,H"„,_iH'„,_2G'„j_3 

 + G,„K,„,_iH"„,_2G'„,_3+ H,„K'„,_iH",„_2G'„j_3 

 + G,„G,i,_iK,„_2G"„,_3 + H„,G',,„_iK,„_2G",„_3 

 H- K„,G",„_iK„,_2G",„_3+ G„,H,„_iK'„,_2G",„_3 

 + H»iII'»!-iK',»-2G",„_3 4- K,„H",„_iK',„_2G",„_3 

 + G,„K„,_iK"i„_2G",„_3 + H„,K',„_iK",„_2Gr"«i-3 



Hence we ohtain the following rule for the determination of : — Write 



down the term G„iG„,_iG,«_2 G„,_j.. We may substitute H and K at 



pleasure for G anywhere except in the last factor, which is always G. 

 Whenever we put H for G, the succeeding letter is to receive a single 

 accent ; whenever K for G, the succeeding letter receives a double accent. 

 The aggregate of all the terms thus formed will be X^, and we may of 

 course obtain similar expressions for fi^, &c. 



Now if we put 



G„i=Z°_,„ X""Y„_,„_3U„_„j_3Zfj_„j_iZ 

 H'„=0, 



G",.= -2XZ°__,Z',_,„ 



H",„=+Z«_„,M,„, 



K"„=-2XZ°__,ZL„, 



