REPORT ON CERTAIN BRANCHES OF ANALYSIS. 247 



minated maxima and minima of the second species, and others 

 also relating to the singular or critical points of curve lines, 

 must depend for their dilucidation upon this more general view 

 of Taylor's series, as connected with the consideration of ge- 

 neral differential coefficients *. 



* Euler has devoted an entire chapter of his Calculus Differentialis to the 

 examination of what he terms the differentials of functions in certain peculiar 

 cases. It is well known that he adopted Leibnitz's original view of the prin- 

 ciples of the differential calculus, and considered diiferentials of the first and 

 higher orders as infinitesimal values oi differences of the first and higher orders. 

 Such a principle necessarily excludes the consideration of differential coefficients 

 as essentially connected with determinate powers of the increment of the inde- 

 pendent variable, which may be said to constitute the essence of Taylor's 

 theorem, and which must be the foundation of all theories of the differential 

 calculus, which make its results depend upon the relation of forms, and not 

 upon the relation of values. As long, however, as the independent variable 

 continues indeterminate, the symbolical values of the differentials are the same 

 upon both hypotheses. But when we come to the consideration of specific va- 

 lues of the independent variable which make differential coefficients above or 

 below a certain order, infinite or zero, then such a view of the nature of dif- 

 ferentials necessarily confounds those of different orders with each other. Thus, 



ii y=. a^ -\- {x — a)^, Euler makes, when a; = a, d y = (d a;)^, instead of 



(5 \ = 3 = (d x)^. Iiy = 2ax — x"-\- a^/ (a^ — a;3), he makes, 

 y) t ^"^ 



when X = a, d y = a \/ — 2 a . d x^, instead of 

 ,j^ ff"^ (d «)■? 



These examples are quite sufficient to make manifest the inadequacy of 

 merely arithmetical views of the principles of the differential calculus to ex- 

 hibit the correct relation which exists between different orders of differentials, 

 and, a fortiori, therefore, between different orders of differential coefficients. 



M. Cauchy, in his Lecons sur le Calcul Infinitesimal (published in 1823), has 

 attempted to conciliate the direct consideration of infinitesimals with the purely 

 algebraical views of the principles of this calculus, which Lagrange first securely 

 established ; and it may be very easily conceded that no attempt of this able 

 analyst, however much at variance with ordinary notions or ordinary practice, 

 would fail from want of a sufficient command over all the resources of analysis. 

 He considers all infinite series as fallacious which are not convergent, and that, 

 consequently, the series of Taylor, when it takes the form of an indefinite series, 

 is not generally true. It is for this reason that he has transferred it from the 

 differential to the integral calculus, and exhibits it as a series with a finite 

 number of terms completed by a definite integral. It is very true that M. Cauchy 

 has perfectly succeeded in dispensing with the consideration of infinite series in 

 the establishment of most of the great principles of the differential and integral 

 calculus ; but I should by no means feel disposed to consider his success in over- 

 coming difficulties which such a course presents as a decisive proof of the expe- 

 diency of following in his footsteps. The fact is, that if the operations of algebra 

 be general, we must necessarily obtain indefinite series, and if the symbols we 

 employ are general likewise, it will be impossible to determine, in most cases, 



