REMARKS ON THE DISTINCTION BETWEEN 



ALGEBRAICAL AND FUNCTIONAL 



EQUATIONS*. 



THE distinction which it is usual to make between alge- 

 braical and functional equations will not, I think, bear a strict 

 examination. It is generally said that an algebraical equation 

 determines the value of an unknown quantity, while a func- 

 tional equation determines the form of an unknown function. 

 But, in reality, the unknown quantity in the former case is a 

 function of the coefficients of the equation, and our object in 

 solving it is simply to ascertain the form of this function. 

 Thus it appears, that in both cases the forms of functions are 

 what we seek. 



Let us therefore consider the subject in a more general 

 manner, and endeavour to find a more decided point of dis- 

 tinction. The science of symbols is conversant with opera- 

 tions, and not with quantities; and an equation, of whatever 

 species, may be defined to be a congeries of operations, known 

 and unknown, equated to the symbol zero. Every operation 

 implies the existence of a base, or something on which the 

 operation is performed in the language of Mr Murphy, a 

 subject. But the base of an operation is often the result of 

 a preceding one. Thus, in log c 2 , the base of the operation 

 log is cc 2 , itself the result of the operation expressed by the 

 index on the base x. This in its turn may be considered as 

 the result of an operation performed on the symbol unity. But 

 in every kind of equation there is a point at which the 

 farther analysis of symbols into operations on certain bases 

 becomes irrelevant ; and thus we are led in every case to recog- 

 nize the existence of ultimate bases. 



* Cambridge Mathematical Journal, No. XIV. Vol. in. p. 92, February, 

 1842. 



