iir 



arbitraries ; and the part of (p.x just mentioned will become 



= Be ' cos X ^ + Be • sin X x ; an expression delivered 

 from the impossible form. But this is a mere artifice of analysis, 

 and does not alter the nature of the function <p. x, which is still 



composed of quantities of the form Be" , though concealed under a 

 certain disguise. It is however well known that cases occur, in which 



quantities of the forms Fc''^' x" , or Vx^,F and Y being 

 arbitrary, form a part of the expression for (p.x ; and, when we 

 consider the heterogeneity of exponential and algebraic functions, it 

 may be perhaps thought not altogether unworthy of our attention 

 to examine in what manner this is brought about. I have en- 

 deavoured in the present paper to give a brief explanation of this 

 difficulty, if such it shall be considered by any. 



The cases alluded to, in which quantities of either of the above- 

 mentioned forms are included in (p.x are those, in which any root 



of the equation, F =: o, is equal to nothing, or in which two or 

 more roots of this equation are equal to each other. First, I shall 

 suppose it to have a root, not actually equal to nothing, but evanes- 

 cent, and represented by A 5 a, li being a finite coefficient. The 

 corresponding term of (p.x will be 



^^ Ax. J» = B.\\ -V hx.la^^h^ X \ ^^ + &c.}, which 

 ultimately is reduced to its first term. It appears then that a 



single root of the equation, V z: o, being equal to nothing will in- 



R 2 



