833 



FUNCTIONS, CALCULUS OF. 



FUNDAMENTAL BASE. 



231 



There was, at one time,'a spirited discussion between Euler, D'Alem- 

 bert, and Lagrange, as 'to whether discontinuous functions could 

 properly be admitted among the solutions of differential equations. 

 We cannot enter into the details of thU discussion, but we shall state 

 the manner in which the question has been settled. 



The considerations in CURVE give some approach to the notion that 

 curves which are perfectly independent may be combined in one equa- 

 tion ; and also that a continuous curve may be drawn, the arc of which 

 runs as near as we please to two distinct branches of two independent 

 curves. The power of expression given by means of definite integrals 

 and periodic series [INTEGRALS, DEFINITE], puts this result in a 

 stronger light. Suppose, for example, that an axis of x being taken, 

 we want to express mathematically the ordinate y, in such manner that 

 it shall be nothing from x = to . = ; that of a certain straight 

 line from x to x = a ; that of a certain circle from x = a to x = b, 

 and nothing again from x = b to x = . In other words, we ask 

 whether a line which is both limited and discontinuous, being a part 

 of a straight line aud a part of a circle, can be put upon the same 

 sort of footing as an algebraic curve ; so that a definite equation y = <f>.f 

 shall always give y = when there is no ordinate, and shall give the 

 proper value of y whenever there is an ordinatc. The answer to 

 this is, that such an expression for <f>x can be given, when the notation 

 of the integral calculus is assumed. And more than this, a continuous 

 curve can be found which fulfils all the conditions above noted, as 

 nearly as we please. That is, m being a small quantity which we may 

 name as small as we! please, it is possible to find a continuous curve, 

 the ordiniite of which shall never be so great as m from x oo to 

 x = 0, shall never differ by so much as m from the ordinate of the 

 straight line while x passes from to a, nor from that of the circle 

 while x passes from o to 6, and finally, shall never be so great as m 

 from x = b to x = o . Having the power of making m as small as 

 we please, we thus trace the expression, as it were, into its final 

 discontinuous form when m = 0. 



In this manner, it appears that every discontinuous function may 

 be regarded as the limiting form of a continuous one; and the 

 various modes in which this is established take away all idea of 

 danger from admitting discontinuous functions among the solution 

 of differential equations. This subject is well studied in its application 

 to phy 



FUNCTIONS, CALCULUS OF. By the term function of a quan- 

 tify is meant any algebraical expression, or other quantity expressed 

 algebraically or not, which depends for its value upon the first. Thus 

 the circumference of a circle is a function of the radius ; the expres- 

 sion (a.- a?) (V*+y) is a function of a, b, x, and y. For the distinctive 

 names of functions, see TRANSCENDENTAL and ALGEBRAICAL. 



All algebra is, in one sense, a calculus of functions ; but the name 

 is peculiarly appropriate, and always given, to that branch of investi- 

 gation in which the form of a function is the thing sought, and not its 

 value in any particular case, nor the conditions under which it may 

 have a particular value. [EQUATIONS, FUNCTIONAL.] For instance, 

 "What is that function of x which, being multiplied by the same 

 function of y, shall give the same function of x + y ? " is a question of 

 the calculus of functions. 



Various isolated questions connected with this calculus have been 

 treated, from the time of Newton downwards, particularly by Lagrange, 

 Laplace, Monge, and Euler. But the direct solution of functional 

 equations, or at least the first attempt to form general methods in the 

 case of functions of a single variable, appears to have been made by 

 Mr. Babbage and Sir J. Herschel (1810-1813). To the treatise entitled 

 ' Examples of the Calculus of Differences,' by the latter, the former 

 appended another, containing examples of the solutions of functional 

 equations. This last, aud the article ' Calculus of Functions,' in the 

 ' Encyclopjcdia Metropolitana,' are the only formal treatises on the sub- 

 ject of which we know. 



A function of x is denoted by <f>x, i|cr, %x, fx, Fx, tx, &c., &c., 

 the first letter being a symbol of an operation to be performed. Thus, 

 Vfx denotes that when the operation signified by / has been performed 

 upon x, that signified by F is performed upon the result. When the 

 same operation is repeated, the results may be denoted by f-r,ffx,fffx, 

 &c., which may be abbreviated into /.<, fx, f*x, &c. For different 

 points of interest connected with the relations of functional forms, see 

 I'KKIODIC ; INVERSE. 



FUNCTIONS, THEORY OF, a name given by Lagrange to a view 

 of the principles of the Differential Calculus, of which we have 

 expressed our opinion in the article DIFFERENTIAL CALCULI'S. The 

 works of Lagrange, in which it details are to be found, are ' Thdorie 

 des ,Fonctions Analytiques,' first edition, 1797, second edition, 1813; 

 and ' Lemons sur le Calcul des Fonctions,' of which the first publi- 

 cation is vol. x. of the ' Lesons de 1'Ecole Normale' (1801); the 

 first separate edition was published in 1797, and the second was pub- 

 lished in 1806. 



Taking Lagrangc's intention to have been the proof that algebra, as 

 it existed in his time, was sufficient to demonstrate the principles nf 

 tin' IMIFureiitiiil Calculus without the introduction of limits, \vn have 

 only to remark that the end is completely attained [I' 

 i ' \ i< i i.i -;.| It is plain to any one acquainted with that calculus, 

 that a dcmoiii-tratiim of Taylor's Theorem being once attained, all 

 the rest follows. We now proceed to look at the proof of this 



theorem given by Lagrange, with reference to absolute correctness or 

 incorrectness. 



Lagrauge first attempts to prove that every function $x has this 

 property, that <p(x + ?t) can be expanded in a series of the form 



<!>(x + A) = <t>x + AA + B/i 2 + c/t 3 



He says, firstly, that no negative powers of h can enter the expansion, 

 for if such were the case <f> (x + 0), instead of being (fix , would be 

 infinite. This is true as to any finite number of negative powers of h, 

 but does not exclude an infinite series of negative powers. For 

 instance, 



1 1. x x- 



A 



_ 



A 3 



_ 

 A 3 



when h = 0,att the terms become infinite, but the first side of the 

 equation is not infinite. Secondly, he assumes that there cannot be 

 fractional powers of h, for if such were the case, there must be frac- 

 tional powers in the original function <j>x, and if <f>x had m different 



p 



values, and if uA* were one of the terms of the development, the re 

 values of this latter, combined with the 111 values of tpx, would give 

 mn different values to <f> (.c + h), instead of m. In answer to this it 

 may be asked how is it known, a priori, that there must be a series of 

 powers of A, every value of which is an expansion of <j> (x + h) '! Hay 

 it not possibly be true that there is an expression of the form 



which is true under certain conditions, determining which of the values 

 of the several terms are to be taken ? Thirdly, he assumes that (having 

 thus obtained a series, in which only whole powers of h are found) the 

 supposition A = must reduce it to its first term; an assumption 

 which can only be admitted of such a series as M + Aft + B/I* + . . . . 

 when it can be made convergent by giving sufficiently small values 

 to A. 



Having once proved or assumed that <f> (x + /i) can be expanded in a 

 series of the form <t>x+Aji + ith*+. . . the proof of Taylor's Theorem, 

 given by Lagrange, does not differ from the common one. He calls A 

 the derived function of </>.*, and denotes it by <p'x : generally, if 

 changing x into x + A change p into p + p'A + ...., p' is the derived 

 function of P. The derived function of Q'JT, denoted by <J>".r, is called 

 the second derived function of 4>.r, and so on. By changing x into 

 x + 1; <t> (x + h), or <px + Alt + sA* + . . . , becomes 



and by changing A into A + k, $ (x + li) becomes 



These must be the same, since both represent </> (x + h + i) : and by 

 equating the terms which contain the first powers of A', we find 



Q'x + VA + B'A" + . . . = A + 2sA + 



whence \=<p'x, 2s = A' = #".!, and so on. The reader will recognise in 

 this process the proof frequently given by means of the preliminary 

 lemma, that if 



_ , ,. ,. dit. du 



ax 



_ 

 all 



The works of Lagrange on this subject, though defective in their 

 fundamental positions, except upon the explanation given in DIFFER- 

 KNTIAL CALCULUS, yet abound in new and useful details, given with all 

 the elegance for which his writings are distinguished : and the student 

 will find them well worth his attention. 



FUNDAMENTAL BASE, in music, is the lowest note of the 

 Perfect Chord, or Triad, as the Germans call it, and of the chord of 

 the 7th : hence it is the root of all real chords ; for chords not derived 

 from either the perfect chord or that of the 7th, arc considered as 

 suspensions or retardations ; or, to speak in unaffected language, the 

 discordant notes of which they are composed are simply afpoyiutm;'.: 

 [CHORD.] 



The following will show the two fundamental Cnordl, and their 

 inversions, with the continued [CONTINUED BASE], or ordinary base, 

 and the Fundamental Rase. 



Fundamental Sate. 



This term is not the best that might have been chosen ; the same 

 meaning is much better conveyed by the word radical, introduced, we 



