89 



Dy - DU =^ CoDuq + CiDui + . . . + c„,iDun-i 

 D^y - D'^ U= CqD^Uo + CiD'^Ui + . . . + c„_iZ)"^m„., 



D"y- Z)" t7= CoD^Uo + c^D^u^ +.., + Cn-xD^u^.^. 



The equation obtained by the elimination of the n con- 

 stants Co, Ci, C2, . . . c„.i, from these last n + l equations, being 

 compared with the proposed equation (1), furnishes us with 

 remarkable results. 



The resulting equation* is 



S (± UoDuiD^Ui . . . D'^'hin-xD^y) 

 = S{± UoDuiD'^u^ . . . D''-'u„^,D"U), 



which, being arranged according to the differential coefficients 

 of y, becomes 



S (±UoDuiD^u.2 . . . Z)"-'M„.2/)"-'Mn-i) D^y 

 -S(± UoDu^D^Ui . . . D^'-hin-iD-^Un-x) D'^-^y + •-• 



...± S{± UoDhiiD'^U2 . . . D^-^Un-iD^Un-i) Dy 

 + S(± DuqD^i . . . Z)"-'m„.2Z)"m„-i) y 

 = S{± UoDuiD^u.2 . . . Z)"-'m„_iD" U). 



Putting this expression, for the sake of brevity, into the 

 form 



SnD'^y - Sn.,D'^-'y+ ...± S,Dy + Soy = S', ] 

 we have the following relations : 



- Sn-\ = A] . Sn (3) 



Sn-2 = A.2-Sn (4) 



* SX+itoDui . . . D"^-'^Un-\D"y) is here used to denote the sum of all 

 the terms derived from u^Dui . . . D^-^Un-iD^y by the permutation of the 

 elements Ug, mi . . . m„-i, y ; each term being regarded as positive or negative 

 according as it may be deduced from that first terra by means [of an odd or 

 even number of interchanges of two letters. 



VOL. IV. I 



