250 ON THE RECENT PROGRESS OF ANALYSIS. 



Mr Talbot ; his mathematical writings bear manifest traces of 

 the ability he has shown in so many branches of science*. 

 But as in this country they seem to have been thought, and 

 by men not apparently unqualified to judge, to contain great 

 additions to our knowledge, I cannot avoid inquiring whether 

 this be true. 



Mr Talbot points out in the early part of his first paper, 

 that if there are n 1 symmetrical relations among the n vari- 

 ables x, y . . . z, then the identical equation 



{y . . . z] dx + (x . . . z) dy + . . . + {xy ...}dz d {xy ... z} 

 will assume the form 



$ (x) dx + $ (y) dy + ...+ <f).(z) dz = d{xy ... z}, 

 and thus give us 



fo (x) dx +j<t> (y} dy+... + J< (z) dz = xy ...z+C. 



Precisely the same remark, though expressed in a different 

 notation, is the foundation of M. Hill's memoir, published in 

 1834, on what he calls ' functiones iteratge.' It will be found in 

 Crelle's Journal, XI. p. 193. A much more general theorem 

 might be established by similar considerations: they are of 

 course applicable whether the function <f> be algebraical or trans- 

 cendent. 



In the course of his researches, Mr Talbot recognised the 

 important principle, that the existence of n I symmetrical 

 algebraical relations among n variables may be expressed by 

 treating them as the roots of an equation, one of whose coeffi- 

 cients at least is variable, the others being either constant or 

 functions of the variable one. Unfortunately he did not pass 

 from hence to the more general view, that the existence of n p 

 symmetrical relations may be expressed in a similar manner if 

 we consider p of the coefficients of the equation as arbitrary 

 quantities. Had he done so, it is possible, though not likely, 

 that he would have rediscovered Abel's theorem ; but as it is, 

 he has never introduced, except once, and then as it were by 

 accident, more than one arbitrary quantity. Thus only one of 



* It must be remembered also that Mr Talbot admits himself to have been anti- 

 cipated to a considerable extent by the publication of Abel's theorem. 



