of certain Functional Equations. 199 



which is impossible. For x ~ being confined to symmetrical 

 Iorms, % [x, *x\ cannot comprehend unsymmetrical forms, nor 

 therefore every function of x, however numerous the forms of 

 X may be; but <p being perfectly unlimited includes both 

 symmetrical and unsymmetrical forms, or, in fact, any func- 

 tional form whatever. Consequently -^-is much more ge- 

 neral than x (****)> an d hence (6) is more general than (7) 

 It seems to me Mr. Herschell has fallen into error in the in 

 ference he has drawn at the end of p. 162 Spence's Essays. 

 Let Px and Q*," says Mr. H., « represent any two parti- 

 cular solutions " of i,x =-t,*x. Then we must have 



Qxz= — QctX. 

 Consequently *L = _£_, and P , = Q*</Ig*£ . and as 



this latter factor is a symmetrical function of *, ux, it appears 

 that the expression Qx. x ~ [x, ax] necessarily includes any 

 other solution as Px." ' 



Now, if my views are right, this inference is much too ae- 

 neral for the very limited condition of the premises. The true 

 definition of a symmetrical function such as y (x, ux) appears 

 to me to be, that the form of the function is variable and 

 utterly independent of the form, relation, or change of the 

 variables; and that these variables simultaneously circulate 

 by a certain substitution made at the same time in each with- 

 out .any_ necessary relation of their values. In the factor 



V ~SrciS however, which is equal to —, neither of these 

 conditions appears to have place. For the form of the 

 function is invariable and dependent on the change of the va- 

 riable, and the variables themselves Px, Qx, if they may be 

 so called, do not change in magnitude or circulate, but sim- 

 ply change their signs; so that the property of a symmetrical 

 function, namely, to retain the same value during the substi- 

 tution, is here preserved by the quantities not changing their 

 values but their signs; which change, too, is neutralized" by the 

 invariable and peculiar form of the function. 



It may be asked, What is the complete solution of (1) ? I 

 have reason to believe that the very simple solution I have 

 given is both a general and complete one, as well as the solu- 

 tions given by Mr. Babbage in the Philosophical Transactions 



ivV 817 ' "' the " ExaIn P les on the Calculus of Finite 



Differences, &c p. 22, or perhaps any other that contains an 

 arbitrary function of x and ax. 



In 



