224 . ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



be SO assigned as that when h was very small, the other two functions should 

 sensibly degenerate to h*. Such a conclusion would probably command the 

 assent of the mind ; but in what sense could this osculation be understood if the 

 functions ^ and ^, expressed throughout their whole extent — series of uncon- 

 nected points — broken portions of lines infinitely small, arranged at infinitely 

 little intervals — denticulated — serpentining in waves such as are presented by 

 what is termed in art a perfectly polished surface — subject to finite changes in 

 infinitely small distances — or, in short, subjected to any of those numerous va- 

 rieties of discontinuity which the modern analysis embraces. So far from the 

 osculation being, in such cases, axiomatic, I believe that few analysts, however 

 skilful, will venture, without much investigation, to assert its existence; and 

 even then, their acquaintance with the infinitely small variations of functions 

 would rather be due to a knowledge of Taylor's Theorem than drawn from the 

 algebraic arrangements which precede it in the calculus. The changes which 

 the second function permits might, indeed, cause it to osculate, or become iden- 

 tical with the third for any given value of x; but still, in this instance also, 

 where the assumption is so far less arbitrary and restricted than Poisson's, it 

 could scarcely be allowed; since, should x be changed, by what principle can 

 we assume that the function ^ h, which osculated with ^ for the former ab- 

 scissa, continues to agree with it for the new point of osculation? 



Mr. Murphy, in one of his excellent memoirs on definite integrals* has 

 given the name of transient functions to a class of expressions analogous to 

 some of those already mentioned. The expression 



(1 — h)( 1 -f-h) 



^ir 



2h (1 — 2t) -^ h^j " 



h = l 



is of this class, vanishing for every value of t excepting that in which t is zero, 

 when the value of the function becomes infinite; and were we to rest the argu- 

 ment merely on functions of this kind, we might very well ask whether it can 

 be at once asserted, as axiomatic, that all the great variety of such functions 

 admit of being developed by powers oft? 



Sir W. Hamilton and Cauchy have both taken the function 



as presenting anomalies and difficulties which cause it to violate two of the 

 received laws of analysis. The last of these authors, indeed, considers this 



» Cambridge Phil. Trans., Vol. V., p. 347. 



