The Interchange of Ttvo Differential Resolvents. 81 



following theorem, viz. : — That if u represent the m th power 

 of any root of the algebraic equation 



a y n + «iy n =\ . . + »„_# + ff B = 0, 



whereof the coefficients a are functions of a single parameter 

 x, then n satisfies a certain linear differential equation which 

 is, in general, of the order ;/. This differential equation 

 admits, when m is a whole number, of a first integration, 

 and may therefore be reduced to an equation of the order 

 n — i ; not, however, always of the same type as the higher 

 equation. And I have shown, more generally, in the same 

 paper, that if 



u = b Q y m + biy m -\ . . + b m _iy + b m , 



where the coefficients b are functions of x only, and m is an 

 integer, we may by known processes, transform the above 

 n — ic equation in y into an « — ic equation in n, and thence 

 derive a linear differential equation of the order n—i, which 

 will be satisfied by any one of the values of u, or by any of 

 the constituents of u. 



5. The object of this paper is to show that if two 

 algebraic equations be so connected as that either can 

 be changed into the other by assuming, without loss of 

 generality, certain relations among the disposable quantities, 

 then cases exist in which the two differential resolvents 

 may also be interchanged by means of the same substitutions. 

 Such interchanges, if practicable, are manifestly important, 

 because they enable us, when one of the differential 

 resolvents is calculated, to determine the form of the 

 other by a simple substitution, and without the labour of an 

 independent calculation. 



6. We have a good example in the following case. The 

 differential resolvents of the two trinomial algebraic 

 equations 



y n - ny + (n - 1)# = (a) 



y n ~ny n -\+(n-\)x = .... (/3) 

 G 



