CEETAIN APPLICATIONS TO THE THEOEY OP DEFINITE INTEGEALS. 751 
those coefficients as functions of the others deemed arbitrary, — or choosing in some other 
way n~r arbitrary elements, express all the coefficients by means of those elements. 
It is further seen that in the one form of the problem the greater the number of 
arbitrary quantities in the equation determining the limits, — in the other form the 
smaller the number of symmetrical equations connecting the limits with each other, — 
the more general is the statement of the problem itself. 
8. The Norwegian and German mathematicians, by whom this branch of analysis has 
been chiefly cultivated, have almost universally followed Abel in his mode of stating 
the general problem, i. e. they have regarded the limits as roots of an equation involving 
a greater or less number of arbitrary elements in its coefficients. Mr. Fox Talbot, in 
his interesting papers “ On the Comparison of Transcendents,” published in the Philoso- 
phical Transactions for the years 1836-37, has in the earlier examples of his method 
expressed by symmetrical equations among the limits the conditions to which the latter 
are subject : in his later examples he adopts the more succinct notation of Abel. Indeed, 
this mode of statement, as it replaces a system of equations by a single equation from 
which such systems may be considered as derived, is far better suited to general investi- 
gations, and will be adopted in this paper. 
9. The complete solution of the problem of the comparison of transcendents, as above 
explained, involves two distinct steps, — 1st, regarding the equation 
E=0 
of (7.), art. 4, as an equation expressing the dependence of the variable x upon the 
variables «!, we must seek to convert the differential expression '2JLdx into a 
complete diflerential relative to «i, «2 • • independent variables, and which will there- 
fore virtually be in the form 
= A, A^cla^ . . fl- K^da,., 
each of the diflerential coefficients Aj, Ag, . . A, being a function of ^i, « 2 ^ • • «r ; 2ndly, we 
must integrate this expression. 
The introduction of the symbol of operation adverted to in art. 1 will enable us to 
dispense with the explicit determination of the coefficients Aj, A 2 , . . A,., and to reduce 
the coiTesponding differential expression to one involving only a single variable. I shall 
now proceed to define the symbol in question, and to investigate its chief properties. 
Definition and Properties of the Symbol 0. 
10. It is an evident consequence of Tayloe’s theorem that we can develope any func- 
tion of x,f{x), in ascending powers of x—a, provided that neither /‘(.r) nor any one of its 
differential coefficients becomes infinite when x=a. To effect this development, we have 
only to assume x—a—z\ then x=a-\-z^ whence 
f{x)=f{a-\-z) 
=/(«)+/(«>+/'(«) i^ + &c. (1.) 
