of this principle itself we commit the error of passing directly from finite to infinite states of functions and variables, instead of estimating and comparing their relations in the different stages of their convergency. I have attempted in what follows to give a rigorous demonstration of the principle in question, at the same time fixing the precise limits of its application, and enumerating the different classes of functions to which it necessarily does not apply. 1. To effect this object I shall begin by explaining what is understood by infinitesimals of different orders. If a function of a converges indefinitely towards zero along with ar, in such a manner that, for a very small value of a, f'(a) shall be less than any given magnitude, the function f(r) at the limit of those values of a which converge indefinitely towards zero is called an infinitely small quantity or infinitesimal. But as for similar decreasing values of a the ratio of convergency may be much higher in one function than in another, we are led naturally to consider indefinitely decreasing quantities of different degrees or orders of convergency. And having fixed upon some one function whose ratio of decrease we assume as the unit of convergency, we call a second function which for similar decreasing values of a decreases in m times as fast a ratio as the first, an indefinitely decreasing quantity of the m” order. We extend this definition to the infinitesimals which are the limits of these quantities, and call the infinitesimal which is the limit of the former of the quantities, an infinitesimal of the first, and the infinitesimal which is the limit of the latter, an infinitesimal of the m” order. Choosing a itself for the function whose ratio of decrease is taken as the unit of convergency, we see clearly that when a is less than unity Aa" is an indefinitely decreasing quantity of the m" order, where m may be integer or fractional. From this we infer that Aa" may represent an indefinitely decreasing quantity of any order, and that the limit of Aa" for values of a which converge indefinitely towards zero may represent an infinitesimal of any order. This we shall designate by the notation lim. -o (Aa"). A very wide generalization, which only suggests itself from the study of the different analytical functions, is given to this definition by defining lim.-, }.f'(a)} to be an infinitesimal of the m” order, if any finite and positive value of m can be found which will render lim...! ?n } a finite quantity. A corollary JC to this definition immediately offers itself, viz. that if lim... } J'(a); is .f'(a) a", an infinitesimal of the m” order, the function increases or decreases indefinitely, while a converges indefinitely towards zero, according as m, and as wo converges towards a finite limit while a converges indefinitely towards zero, it depends upon the sign of m, -m whether Z.() increases or decreases indefinitely at the same time. But if no finite and positive value of m can be found which will render lim...{ tity, there are two cases to be considered. 1st. If wo converges inde finitely towards zero along with a however great m may be taken, it follows from the general definition of an infinitesimal of the m” order that lim.-, }.f'(a)} is an infinitesimal of an infinitely high order. 2d. If Vo increases indefinitely towards ; for values of a which converge in definitely towards zero, however small m may be taken, it follows from the same general definition that lim --, }.f'(a)} is an infinitesimal of an infinitely low order. 2. Of infinitesimals in general I may enunciate the following theorem. THEOREM. If lim ,-, } f(a)} is an infinitesimal of the m” order, and if lim.-6 p(x)} is an infinitesimal of the mo" order, the equation f(r) = p(r) cannot exist for the general value of a. DEM. For if f'(a) can be equal to p (a) for the general values of ar, dividing by a ", we find that wo can be equal to *g). which is equal to ‘b (a) - ––. Now by hypothesis 49 and Po) converge a;" 1 a" - mi a" towards a finite limit as a converges indefinitely towards zero, whilst I converges towards ; or zero at the same time, according as m, is greater or less than m. Therefore a quantity converging towards a finite limit can throughout be equal to another which converges towards ; or zero, which is absurd. Therefore f'(a) cannot be equal to Hence we see that the law of homogeneity, which is so essential an element of all analytical developements, holds good at the limits of the functions and variables as well as for values varying between finite limits. We shall now see that this law is alone sufficient to demonstrate La Grange's principle within the proper limits of its application, as well as to indicate at once the cases in which it is necessarily inapplicable. 3. I shall now enunciate and demonstrate La Grange's principle. THEOREM. If f(a) be a function of a continuous between the limits 0 and all, and if lim.-, }.f(a)} is an infinitesimal of a finite and positive order oc, the function f(a) for any value of a within those limits may be analytically represented by a series of terms of the forms Aw" + Baft + Cay-H &c. where A, B, C are finite coefficients, and a, 3, y finite and positive exponents. DEM. As lim -, }.f'(x)} is an infinitesimal of the finite and positive order sc, it follows that the limit of the ratio wo for the values of a which converge indefinitely towards zero, is equal to a finite quantity. This finite quantity is the coefficient A. From this and from the continuity of the functions (the difference of two continuous functions being itself a continuous function) it follows that if we make a increase insensibly from zero to some finite quantity within the limit ar, the values which the functions for any finite value of a within the limit ar, f'(r) may be analytically represented by the series Aa" + Ba'4. Cao + Dr” +&c. Q. E. D. CoR. We may from the preceding proposition deduce a mode of finding successively the terms Aa", Bao, Cao, and thus of actually effecting the developement of f(r). This is best explained by an example. Let f(a) be sin a and assume sin a = A a 4- Baft + Cao +&c. Dividing Sill a’ verge indefinitely towards zero, as 3 > a. and y > a, it is manifest that A is equal to lim. -o (**) where a is that finite and positive number sin - - - which can render lim...( rules of the Differential Calculus for finding the values of fractions which - - 0 for certain values of the variable become 0, we find that which for a = 1 becomes finite and equal to unity. Therefore Treating the function sin a – a in the same manner as we have just -*-* –o1.2.3 * 1.2.3.4.5 This process is general, and may be easily applied to demonstrate the theorems of Taylor and Maclaurin. 4. The theorem of the last article is La Grange's principle, and was used by that analyst as the fundamental principle of his Calculus of Functions. By the corollary to the theorem in Article 2, it is clear that it is inapplicable to functions whose limits for the values of a which converge indefinitely to zero are infinitesimals of an infinitely high or an infinitely low order, Aa", Bao, Cao being at the same limit infinitesimals of finite orders. There are however only two known classes l of functions which have this property, viz. e o and The limit of I log as the ratio of e- to a " is easily shown to be infinitely small, however great a may be taken. Lim ...( ) is therefore an infinitesimal of an infinitely high order, and consequently • * cannot be represented by a series like Aa" + Bao 4-Cay 4. &c. On the contrary, the limit of the ratio 1 of loga. to a " may be shown to be infinitely great, however small a may be taken. Lim. -o o is therefore an infinitesimal of an infi nitely low order, and therefore cannot be represented by a series such as Aa" + Bar” + Car? -- &c. In either case indeed, if we assumed the principle, we should, by passing to the limits of the equivalent series, find an infinitesimal of an infinitely high or infinitely low order, equal to a series of infinitesimals of finite orders, which would violate the principle of homogeneity which exists equally in finite and infinitesimal quantities. |