232 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, , 



log (X — n) 



invented for this purpose. As o'', a™, o", or e ''"^"^ , which last has the 



properties assigned to A. 



Our equation (6,) in short, by whatever signs of mere agreement we express 

 the coefficients A and B, amounts really to the two equations 



X X 



{y = log x} 



a. 



n 



f y = sin x| 



n 



and has no other advantage than results from uniting these two equations in 

 one. 



Simple expressions connecting, with the ordinary arrangements of algebra, 

 the different orders and kinds of hredks to which continuity is subject, are still 

 wanted ; expressions that should perform their work as simply as the various 

 series and formulse used for interpolation. Those functions to which Mr. Mur- 

 phy has given the name of transient functions serve well enough for most of 

 the cases where such connexion is not wanted. Of these transient functions, 



expressions of the class 



x — a . 



a — a 



are the most simple ; and I presume that most analysts have long used them to 

 express such breaks of continuity as are represented in the function 



d) X = f X H f X + F X. 



X — a X — b 



But, notwithstanding the necessity which thus exists for retaining expres- 

 sions that shall possess, independently of agreement, the properties in question, 

 it seems to me that advantage would accrue from uniting these methods ; and 

 that not only some convenient notation should be agreed upon to represent 

 these semi-constants that change per saltum at limits, but that the simplest 

 class of periodic variables should be dealt with in the same manner, and made 

 types of reference to which all periodic quantities could be reduced. 



With this view I have elsewhere recommended that any given function F x 

 should be made periodic between the limits 1 and 'i by annexing the symbol 

 '^ ; so that 



-©.Ftx) 

 would represent, geometrically, a series of waves formed by the continued re- 

 petition of the same portion of the given function. 



