572 Mr. G. B. Jerrard's Reflections on the Resolution 

 And combining this equation with 



W/ + D, W/ + . . + Dfi = 0, 

 we shall ultimately obtain 



*R also being expressive of a rational function. 



The equation for W will therefore belong to a class of 

 equations of the sixth degree, the resolution of which can, as 

 Abelt has shown, be effected by means of equations of the 

 second and third degrees. 



Whence I infer the possibility of solving by a finite com- 

 bination of radicals and rational functions the general equa- 

 tion of the fifth degree. (31.) 



44^. The equation of which 



Wf +^R(Wy") 



is a root will evidently be of the third degree. For omitting 

 the parentheses connected with ^R, we see that 

 ^RW^>='R'^W^(^.-> = W^<(,.), 



the exponent, as is usual, indicating a repetition of an opera- 

 tion ; and that consequently the root in question will not be 

 affected by writing/' (^ e^ instead of/\ 



We must also have 



( W^. + m W^-) (a b) (c d) . . = ( Vp) (a b) (c d) . . , 



when (ab) (^cd^ . . takes the form (aja) (ab); but not for 

 all values of a, b, c, d, . . : since the method of continuous 

 substitutions will not generally be applicable to processes 

 based upon the theorem (v, w.), which is, as we must remem- 

 ber, hypothetical in itself. 



Hence I conclude that there will be an equation of the third 

 degree with given coefficients simultaneous with the equation 

 V^'^ + Cj V14 -f. . . =: 0, which cannot be depressed below the 

 15th degree without inducing certain relations among Aj, Ag, 

 .. A4. 



* We shall have, as before, 



Wy.«=lR(lR(Wy.-)). 



f In a memoir " Sur une classe particuliere d'Equations resoliibles alge- 

 briquement." Crelle's Journal, vol. iv. p. 131. 



I All this will be more readily understood from considering that, in ex- 

 panding any function whatever of j:'+// two independent quantities, we do 

 not necessarily obtain an expression symmetric with respect to x and A, and 

 such as to admit of an interchange between them. In fact, instead of the 



