89 



Dy - DU = CoDuo + c,Z)M| + . . . + c„.i£)m„., 

 D'y - D* U= CoD^Uo + c.D^m, + . . . + c„.iD'u„., 



D"ij- D''U= CoD"Uo + CiD"u^ + . . . + c„.jD''w„.,. 



The equation obtained by the elimination of the n con- 

 stants Co, c„ Ci, . . . c„.i, from these last n + 1 equations, being 

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

 remarkable results. 



The resulting equation* is 



5 (± ttoDuiD^Ui . . . D^-^Un-xD^y) 

 = S{± UoDuiD'^Ui . . . D^WiD^U), 



which, being arranged according to the differential coefficients 

 of y, becomes 



S(±UoDuiD^Ui . . . Z)'»-2m„.2Z)»-'m„.,) D'y 

 -S(± UoDuiD^Ui . . . Z)"-2«„.2/)"M„.,) Z)»-V + . . . 



...±S(± UoD^UiD'ui . . . D''-'u„.2D"u„.,) Dy 

 + s\± DuoD^Ui . . . £)"-'m„.2D"«»-0 y 

 = 5 (± UoDuiD^Ui . . . D^-'un.iD" U). 



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

 form 



SnD^y - Sn-xD^-'y^ • . . ± SyDy + ^o^ = S', ] 



we have the following relations : 



- <Sn-l = A].Sn (3) 



Sn-2 = -^a- Sn ('*) 



' S(+uoD«i . . . D"^i(H-\D"ti) is here used to denote th« sum of all 

 the terms derived from uoDui . . . X)" 'i/„-i£)''y by the permutation of the 

 elements uo, «i . . . u„ i, y ; each term being regarded as positive or negative 

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

 even number of interchanges of two letters. 



VOL. IV. I 



