LOUIS CLARK VANUXEM FOUNDATION 
 
 THE THEORY OF 
 PERMUTABLE FUNCTIONS 
 
 BY 
 
 VITO VOLTERRA 
 
 Professor of Mathematical Physics in the 
 University of Rome 
 
 Lectures Delivered at Princeton University 
 October, 1912 
 
 PRINCETON UNIVERSITY PRESS 
 
 PRINCETON 
 
 LONDON : HUMPHREY MILFORD 
 
 OXFORD UNIVERSITY PRESS 
 
 I9I5 
 
Published April, 191 5 
 
LECTURE I 
 
 302503 
 
LECTURE I 
 
 1. We shall begin with quite elementary and 
 general notions. 
 
 First, let us recall the properties of a sum 
 
 n 
 
 «1 + ^^2 + • • • + ((n = ^i ^i • 
 
 I 
 
 This operation is both associative and com- 
 mutative, that is, 
 
 [a + b) + c = a+(b-i-c) 
 and 
 
 a -\r h =^ h + a . 
 
 Now we can jDass from a sum to an integral 
 by a well-known limiting process. For the 
 sake of simplicity, we shall make use of the 
 definition of Riemann: Given a function f (<.v) 
 which is defined over an interval ah, we sub- 
 divide the interval ah into n parts //i, h-, hs, 
 h^ hn . Corresponding to every in- 
 terval hi we then take some value /, of /(<r) 
 lying between the upper and lower limits of 
 /(.r) on hj , and we form the sum 
 
 i /; ^. . 
 1 
 
4 THEORY OF PERMUTABLE FUNCTIONS 
 
 Now suppose we allow hi, /lo, Jh, h ,( to 
 
 become indefinitely small. Then if a unique 
 limit is approached by the sum regardless of 
 the way in which the subdivision of ab is made, 
 we have 
 
 lim tifihi= I f{x)dx . 
 
 I */ a 
 
 Necessary and sufficient conditions for the 
 existence of this limit are well known. In 
 particular, if the function j{oc) is continuous 
 over the interval ah or has at most a finite 
 nmnber of discontinuities, the limit and hence 
 also the integral exists. 
 
 2. Now let us form the product 
 
 a ■ h c . 
 
 This operation is associative and commutative, 
 that is to say, 
 
 {ah) c ^^ a [he) 
 and 
 
 ah "= ha . 
 
 It is not worth om* while to consider the oper- 
 ation which could be obtained from a product 
 by a limiting process such as the one employed 
 
LECTURE I 
 
 in defining an integral. We should be led to 
 logarithmic integration. 
 
 3. However, let us consider a limiting pro- 
 cess which leads us to something more than 
 these elementary operations. 
 
 Let us choose a set of numbers 
 
 m, 
 
 where 
 
 i,s^ 
 
 -1,2, 
 
 
 • 9 
 
 an array 
 
 
 
 
 
 mn 
 
 mi2 
 
 
 
 w^2l 
 
 ^22 
 
 
 
 Mg^ 
 
 nigz 
 
 and 
 
 numbers 
 
 nis. 
 
 that 
 
 is, 
 
 
 
 
 
 nn 
 
 nn 
 
 
 
 n2i 
 
 ^22 
 
 which may be written in 
 
 m 
 
 Iff 
 
 7)h 
 
 Ig 
 
 m 
 
 99 
 
 where /, 6= 1, 2, 
 
 9 
 
 *ffi 
 
 'ff2 
 
 ''^9 
 
 ^20 
 
 We then consider the operation 
 
 9 
 
 (1) S„ riUj, nf,^ 
 1 
 
 which we shall call composition of the second 
 type. This operation is associative, for if we 
 
6 THEORY OF PERM LIABLE EL'XCTIOXS 
 
 also introduce a set of numbers jhsi where 
 
 2, s = 1, 2 . . (J , and write the sum 
 
 y y 
 
 X,, %k m^i, n,,k J)/,, 
 
 1 1 
 
 the expression which we thus obtain is equiv- 
 alent to either of the forms 
 
 .'/ y 
 
 S/, (2;, lUif, n,„) J),,, 
 
 1 1 
 
 9 y 
 
 2/, ntif, (2^. ti;,/, j)ks) 
 1 1 
 
 which f)i'oves that the associative law is sat- 
 isfied. 
 
 INIaking use of the notation 
 
 1 
 we shall have 
 
 which may be written without the parenthesis, 
 thus, 
 
 The commutative law will in general not 
 be satisfied. When it is, the quantities under 
 consideration are called permutable, and we 
 have 
 
LECTURE I 
 
 {m, n)i, = {n, m)i,. . 
 
 We can at once give an example involving 
 permutable quantities. All that is necessary 
 is to consider 
 
 {m, m)ir which nia}^ be written (m^)ir 
 [m, m, m)ir which may be written {ni^)ir 
 and so on. And it is clear that 
 
 since the associative law is satisfied. 
 
 4. We shall consider also another operation 
 similar to the last, namely 
 
 (21 
 
 s— 1 
 
 which will be called composition of the first 
 type. The smn (1) previously considered re- 
 duces to this one if we suppose that the num- 
 bers are zero unless the second subscript is 
 greater than the first. In other words, we 
 have in this case 
 
 
 muy m^ 
 
 n-i 
 
 
 
 7)1 
 
 23 
 
 
 
 
 m 
 
 f/-i,y 
 
8 THEORY OF PERMUTABLE FUNCTIONS 
 
 Let US represent the sum (2) by 
 
 This expression vanishes if s is less than or 
 equal to / + 1. Moreover, if we write 
 
 we shall have 
 
 which vanishes if s is less than or equal to 
 I + 2, and so on. 
 
 In general, it is not true that 
 
 but when this condition is satisfied, the two 
 quantities are called permutable. To distin- 
 guish this sort of permutability from that 
 which we defined in section 3, we shall say that 
 the new and the old are of types one and two 
 respectively. In other words, if 
 
 g g 
 
 (8) S;, ;;/,;, nj,g = 2,, n^, m,„j 
 1 1 
 
 we have permutability of tlie second type, 
 
 whereas if 
 
 s-l s—\ 
 
 i+l i+1 
 
LECTURE I 9 
 
 we have permutability of the first type. 
 Clearly, if we put 
 
 we at once obtain an example of permutability 
 of the first type like the one mentioned in the 
 last section. Nevertheless, we shall give an- 
 other example which is of interest. Let us 
 suppose that n^^ = 1. Then the condition for 
 permutability becomes 
 
 .s— 1 S-1 
 
 Putting s =^ I -\- 2, \\Q have 
 
 and putting .s ^ ^ + 3, we have 
 
 whence 
 
 and so on. Owing to the above, we have 
 
 for all values of r, s, and g. From this it fol- 
 lows that the matrix of the ins is of the follow- 
 ing type: 
 
10 
 
 THEORY OF PERMUTABLE FUNCTIONS 
 
 ^^1 i(.> ((.^ 
 
 U U ((i (ir, 
 
 U U f^i 
 
 'y-1 
 
 ^g-i 
 
 U u u . 
 
 U . . . u . 
 The law for this matrix may he expressed hy 
 
 Tih, = ;«,_,• , 
 which puts into evidence the fact that the val- 
 ue of m depends upon the difference between 
 the two subscripts 8 and i. 
 
 5. Now what do we find when we pass to 
 the limit by a process analogous to tlie one 
 employed in the integral calculus in going 
 from a sum to an integral? We there passed 
 
 from a set of quantities /i, f-^, /a, f,, 
 
 with single subscripts to a function f{d') of 
 one independent variable aj, the variable taking 
 tlie place of the subscripts. Here, we have in- 
 stead a set of quantities ;«,.^ with double sub- 
 scripts; hence, we must replace them by a 
 function of two variables 
 
 /G^,.y) 
 
LECTURE I 11 
 
 where the two variables take the place of the 
 double subscripts. Moreover, we also have 
 another set of quantities „ .^ which must be re- 
 placed by a different function of two variables 
 (l>{:v,^). Finally, 2/1 %7i W;,s must be replaced by 
 
 J/(.r, (/>(£, ^)r/£. 
 We thus obtain two operations: 
 Composition of the first type: 
 
 Composition of the second type: 
 
 ^ a 
 
 The condition for permutability of the first 
 type is 
 
 ff[x, ^) <^(e, //) d^ =f'^(x, i) M, y) dt. 
 
 for permutabilitj" of the second type, 
 
 ff{x,^.)<^{^,U)d^=Pi>[x,^)f[^,y) d^^ . 
 
 a a 
 
 The associative property is always satisfied. 
 6. Let us begin by examining permuta- 
 
12 THEORY OF PERMUTABLE FUNCTIONS 
 
 bility of the first type. The most miportant 
 facts are here summarized: 
 
 1. All of the functions which can be ob- 
 tained by the composition of permutable func- 
 tions are permutable with one another and 
 also with the original functions. 
 
 2. All of the functions which can be ob- 
 tained by the addition or the subtraction of 
 permutable functions are permutable with one 
 another and with the original functions. 
 
 Now, let us see how the following problem 
 may be solved: To determine all the func- 
 tions whicli are permutable with unity. 
 
 We can readily solve this problem if we re- 
 call a question whicli has already been 
 answered. For before passing to the limit, we 
 saw that if the functions ?« ., were permutable 
 with unity, the condition 
 
 was satisfied. Now since, in the limit, the sub- 
 scripts are replaced by the variables 07 and y, 
 we are led to infer that 
 
 This we can prove immediately. For if 
 
LECTURE I 13 
 
 ^ X X 
 
 it must follow that 
 
 d (h 3 6 .^ ^ 
 
 Hence ^ and / are of the forms ^{y — x) and 
 j[y — x) respectively. 
 
 Moreover, all of the fmictions of the type 
 /(?/ — x) are permutable with one another; for 
 
 as can be verified at once. The functions of 
 type f(y — x) form a group of permutable 
 functions which is of especial interest. We 
 have called it the group of closed cycle.* How- 
 ever, we shall not go into an examination of 
 it here. 
 
 7. We have used several different notations 
 representing the operation of composition. 
 The simplest scheme where no confusion with 
 multiplication is liable to arise, is merely to 
 write 
 
 f(f> or f(f>ix,?/) 
 
 *Le5ons sur les equations integrales et int6gro-differentielles. 
 Paris: Gauthier-Villars. 1913. P. 150. 
 
14 THEORY OF PERMLTABLE FUNCTIONS 
 
 to rej^resent the resultant of the composition 
 of two function / and (f) . But in a case where 
 confusion might arise, we may place a small 
 star over the letters, thus 
 
 f4.. 
 
 We may also put the letters in square brackets 
 
 [/■■^]- 
 
 just as in the above we wrote [;;/. 71']/,,, to rep- 
 resent the composition of the quantities niij^ 
 and mfj, . 
 
 To indicate the composition of / with itself, 
 the composition of the resultant thus obtained 
 with f, and so on, we shall write 
 
 r-,/', .... 
 
 respectively, and if no confusion with multi- 
 plication is liable to arise, we may even omit 
 the small stars and write 
 
 J ■> J 1 
 
 8. The notation in certain cases demands 
 particular examination. Thus, to indicate the 
 product of a constant a by /, we write «/; and 
 if h is also a constant and 4) a function which is 
 
LECTURE I 15 
 
 permutable with f, then 6 ^ is permutable with 
 af and the composition of the two gives us 
 
 « » 
 a h f ^ . 
 
 Moreover, if we have two polynomials made 
 up of permutable functions with constant co- 
 efficients, these polynomials will also be pre- 
 mutable with one another, and to effect their 
 composition all that is necessary is to apply 
 the same rule which is used when polynomials 
 are multiplied together. 
 
 But if a and h are constants, then a + / 
 and & + will not in general be permutable 
 with one another. Nevertheless, we shall ex- 
 tend the definition so as to have in this case 
 
 (a+f) (b-{-4) = ab + «(^ + hf + f^ . 
 
 9. Before going further, let us consider 
 what takes place in the case of composition 
 of the second type. 
 
 All that we have said above concerning per- 
 mutability of the first type can be established 
 for permutability of the second type barring 
 the remarks on permutability with unity. 
 With this exception, all of the properties just 
 mentioned may be extended to this case at once. 
 
16 THEORY OF PERMUTABLE FUNCTIONS 
 
 10. Let US pass in review some of the most 
 interesting properties which can be derived 
 from the operation of composition. We 
 shall return once more to the finite case and 
 consider the operation of composition for the 
 numbers viig. We saw above that if we com- 
 posed the w^s with the n's, the resultant thus 
 obtained with the p^'s and so on, then after 
 s-i-1 compositions, the resultant would be zero. 
 In a like manner, all of the symbolic powers 
 of 7Jiig beginning with [7}f~^^^~\i, have a value 
 zero. 
 
 Having seen this, let us consider any ana- 
 lytic function 
 
 00 
 
 
 which converges within a certain circle, and 
 let us write 
 
 
 
 This new expression is evidently an integral 
 rational function of z, that is, a polynomial. 
 
 Moreover, this result may be generalized. 
 Consider the analytic function 
 
 00 00 
 
 ^ 
 
LECTURE I 17 
 
 of more than one variable, and write 
 
 
 
 *1 ^2 . . . 2g <= . 
 
 We again obtain a polynomial. 
 
 11. Now, what does the above theorem 
 become when we pass by a limiting process to 
 the composition of functions? This we pro- 
 ceed to investigate. 
 
 Consider 
 
 
 
 We shall prove that if f (x, y) is finite „ this 
 eocpression is always an entire function of z, 
 whatever may he the absolute value o/ f (x, y) . 
 To prove this theorem, we notice that 
 
 where M is some finite quantity and where It 
 is less than the radius of convergence of the 
 series. Moreover, let /u, be a quantity which 
 is larger than the absolute value of /. Then 
 we shall have 
 
18 THEORY OF PERMUTABLE FUNCTIONS 
 
 2! 
 
 \J\^,i/)\< — 31 — • • • • 
 
 and so on, whence 
 
 which proves that the series is convergent for 
 all values of z and is thus an entire function 
 of %. 
 
 This theorem may also be generalized. Let 
 us consider the series 
 
 00 00 
 
 Sh . . . Si, y4i . i Z^' . . . Z,'e 
 
 ^ " 
 
 which represents the expansion of a function 
 F [z^, Z2, 23, . . . .Ze) about a point, and which 
 converges if the absolute values of Zi, Zo, S3, 
 . . . .Zg do not exceed certain limits. Then 
 the series 
 
 0^ 
 
 is always an entire function of z^, z^, Z2,. . .s„. 
 
LECTURE I 19 
 
 The proof is made in the previous case. Thus, 
 if fi> f-ij fs,' ' ■ -fe are each less than /a in abso- 
 lute value, then 
 
 uh+.--+ie f^ — M... +»,_! 
 fn fi.\ < —^ ^ ^ — 
 
 l-^i ••••^'''l^(z\+4+ ... +4-i)i 
 
 and hence the theorem may be verified im- 
 mediately. We may also demonstrate another 
 property besides the one just shown. Indeed, 
 we have up to the present regarded the func- 
 tion F (zi,. . . .z^, \ oj, y) as Si function of Zi, Z2, 
 Zs, S4, .... Zp, but it is also a f miction of cc 
 and y. Regarded as a function of these two 
 variables, the function is permutable with the 
 functions /i .... fe. This may be seen at 
 once; for owing to the uniform convergence 
 of the series, the operation of composition 
 may be performed term by term, and since 
 each term of the series is permutable with /i, 
 /2> fs, ..../(., so also must be the sum. 
 
 To sum up, we have the following theorem: 
 
 If 
 
 22 . . . XAi 
 
 i^ *1 
 
 n 
 
 is the expansion of aii analytic function about 
 a point, then 
 
20 THEORY OF PERMUTABLE FUNXTIONS 
 
 where f i, io, f 3, . • . . f e are permutable func- 
 tions, is an entire function of Zi, z^, Z3, . . . .Ze, 
 aw6^ a* a function of x aw^Z y is ijermutahle 
 with the functions fi, £2, fa .... fe. 
 
 Now if in F [z^, z^,. . . .z^ \ oc, y) we put 
 Si = 2^2 = ~3 = .... Sg = 1, we obtain a series 
 
 which is convergent for all values of the fs. 
 
 12. The theorems which we have been de- 
 riving above suggest a method for investigat- 
 ing to a considerable extent the j^roperties of 
 permutable functions and for carrying out the 
 operations of composition. 
 
 Thus, let us consider any analytic expression 
 
 F {,,...,,) 
 
 which can be expanded about the point z^ = 
 ^2 — S3 = 2:4 = .... Ze = in powers of Zi, 
 Zo, z-i, . . . . Zg. If we replace z^, Zo, z^, .... Zg 
 in the series by /i, /2, /s, • • • • /. respectively, 
 and write the operation of composition wher- 
 ever we previously had multiplication, we shall 
 
LECTURE I 31 
 
 always obtain a series which converges for all 
 values of /i, /a, fs, .... fe, and which repre- 
 sents a function permutable with fi, f2, f-i, .... 
 fe . We may represent it by 
 
 Thus, every algebraic expression takes on a 
 new meaning for the operation of composition. 
 For example, 
 
 — y ..,2 _l_ ^3 
 
 is a series which converges within the unit cir- 
 cle. But if we write 
 
 J • • • 
 
 1 + / -^ ^ '' 
 we obtain a series which converges for all 
 values of / and which is permutable with /. 
 Consequently, a meaning has been ascribed to 
 the expression on the left hand side of the 
 equation. 
 
 Moreover, if we take two expressions 
 
 
 and 
 
22 THEORY OF PERMUTABLE FUNCTIONS 
 
 ~, i.= / + /'+/'+ • . . 
 
 1-/ 
 
 then to make the composition of the two left- 
 hand members, it is only necessary to apply 
 the rules for finding their algebraic product 
 and we shall have 
 
 . = /■2+/^+ . . . 
 i-f ^ ^ 
 
 Hence, all the rules of ordinary algebra re- 
 main valid when we pass from the field of mul- 
 tiplication to the field of composition. 
 
 Some of the consequences which can be de- 
 rived from this fact will be seen shortly. 
 
 13. Now let us see what takes place for 
 the second type of composition. 
 Let 
 
 be the ratio of two entire functions ^{p^) and 
 1 + i// (2;) which are such that </)(0) = v/'(0) 
 = 0. 
 
 Then we shall have an expansion 
 
LECTURE I S3 
 
 F{z) = A,s + A.,z^ + A^z^+ ... 
 which converges in general within a certain 
 circle having the point ;:; = as center. 
 Now let us consider the expression 
 FU^-) = A,m,,z + A,(m%z^+... 
 We shall prove that this new series is the quo- 
 tient of two entire functions. 
 
 14. Let us first write 
 
 <t>(z) = B,z + Bo .r + . . . 
 and determine a quantit}^ 
 
 ^is{^) = ^1 ^^hs •«• + B2(m% z^+ . . . 
 We say that ^j, (z) is an entire function of z. 
 For let ju, be greater in absolute value than m^s . 
 We then have (§3) 
 
 I w^,,| <(i, I {m'% \<g li\ \ {ni^)is !</>%... 
 Moreover, 
 
 1^1 \iiz + \B, \^'gz' + 1^3 \^r^+ ••• 
 converges for all values of z, and the theorem 
 is proved. 
 
 For the same reason, if 
 
 .\,(,) = C,,+ C,^+ ... 
 then the series 
 
 x\fi,{^) = C^ mt, .*• -V C\{7n%, £^ + ... 
 is an entire fmiction. 
 
24 THEORY OF PERMUTABLE FUNCTIONS 
 
 Bearing these facts in mind, let us consider 
 the system of algebraic linear equations 
 
 g 
 
 1 
 If we replace the unknowns X,^ by F^^ we 
 can verify without trouble that these equa- 
 tions are identically satisfied. But if we solve 
 for the unknowns X,, in the above system, the 
 solution will be expressed as the quotients 
 of rational entire functions of i//,^ and <^j, and 
 hence the quantities X^^ are quotients of entire 
 functions of z. 
 
 It is clear that the determinant which con- 
 stitutes the denominator of these quotients 
 cannot vanish identically, and hence the theo- 
 rem is 2)roved. 
 
 It is not difficult to generalize this. Thus 
 instead of the quotient (•!), let us write 
 
 where (f) and xjj are entire functions of the 
 variables Zi, Z2, Zs, .... Zg which vanish for 
 
 Zi Z2 S3 , . . . Zc U. xl 
 
 is the expansion of F about the point for which 
 
LECTURE I 25 
 
 Zi = Z2 = ^z = .... = Ze = 0, and if we write 
 
 F,,{z, ,...z,) = t.. .tA,^ . . . ,^(nf^n'"- . . . q'^) z^\ . .^/a 
 
 where m , n , .... q are permutable, then 
 the function F (si, S2, S3, . . . . s^ | .r, ?/) will be 
 the quotient of two entire functions of Z\, z^, 
 
 Zz, .... Ze. 
 
 To make the proof in this case, it is only 
 necessary to repeat the argument given above. 
 We may add that Fi^ is permutable with m^, 
 
 "is J • • • • ^ts • 
 
 15. Let us now pass to the limit, that is, 
 let us consider permutable functions of the 
 second type*. All of the theorems remain 
 valid. In other words, we have the theorem 
 that 
 F(,,, ...^ x,y) = t... tA,^ . . ,;^/V> . . .f:^ zt . • . ^;', 
 
 where j\, J2, jz^ - - - - /« are permutable func- 
 tions of the second type, is the quotient of two 
 entire functions. Also, 
 
 F[z^,...z,\x,y) 
 
 * To indicate composition of the second type, we shall make 
 use of a double star thus: 
 
 •* ** 
 /l 1\ 
 
26 THEORY OF PERMUTABLE FUNCTIONS 
 
 is a function which is permutable with fi, f2, 
 
 U, ■■■■ fe. 
 
 We shall study some applications of these 
 fundamental theorems in the second lecture. 
 
LECTURE II 
 
LECTURE II 
 
 1. We shall begin by classifying integral 
 equations into several categories. First, let 
 us examine those which are linear. The sim- 
 plest ones which we rmi across are the 
 following : 
 
 (1) / W + /oVW F[x,y) dx = ^y) , 
 
 known as Volterras equation of the second 
 kmd, and 
 
 (10 f{y)-^§\f(x)F{x,y)dx = <l>(y), 
 
 Fredholm's equation of the second hind. 
 We shall also consider certain other kinds fur- 
 ther on. 
 
 Let us look at equation (1). If we multi- 
 ply both sides by ^ (y, z) and integrate with 
 respect to y between the limits and z, we 
 obtain 
 
 jl^(y,z)fiy)dy-^ ^]f(x)dx jy(x,y)^(y,z)dy 
 
 = ^\^[jj)^{y.^)dy . 
 
30 THEORY OF PERMUTABLE FUNCTIONS 
 
 If now the function ^ be so chosen that 
 
 (A) j[F{x,i/) $ [i/, z) dy = - <^[x, ,) - F(x, z), 
 it will follow that 
 
 / ^ (^' ^) f^y) '^y~ ^\ *(-^^' ^) /(^) ^^^ 
 
 - /J F{x, z) f{x) dx = f'jiy) <D(y, z) dy 
 which is reducible by (1) to 
 
 /(.•) = <^(.~)+/V(^)(D(^,,.)./y, 
 
 ^ 
 
 so that the difficulty has been narrowed dowTi 
 to the solution of (A). In the symbols for 
 composition of the first kind, this equation 
 maj^ be written as follows: 
 
 (2) ^{x.y)-^ F{x,y)-\- F6{x,y) = ^ . 
 If we apply a similar argument to equation 
 (1') we find that the solution will be given in 
 the form 
 
 f{z)=^{z)^ f^{y)^[y,z)dy 
 ^ 
 
 where 
 
 (2^) ^{x,y) + F(x,y) + F*i(x,y) = . 
 
 2. Let us first see how equation (2) may 
 be solved. If we write 
 

 LECTURE II 
 
 
 
 z 
 
 ? + 2"! + Z Zi — 
 
 
 
 
 we shall obtain 
 
 
 
 
 — z 
 
 = -. + /- 
 
 _ -3 
 
 <5 
 
 + 
 
 "^~1 + ^ 
 
 the solution heir 
 
 [g valid if z 
 
 < 
 
 1. 
 
 But suppose 
 
 we write 
 
 
 
 _ ^ _ 
 
 = -F+ F^- 
 
 . j^3 
 
 + 
 
 31 
 
 1 + F 
 Then, in this case, we have an expansion which 
 converges for all values of F and which satis- 
 fies equation (2) by what we have proved. 
 Hence we shall have 
 
 $= -F+ F''-F^+ . . . 
 and the integral equation (2) is solved. 
 
 If we replace F (oD,y) hj u F (x,y) in equa- 
 tion (2) we obtain 
 
 ^{x,f/) + u F(x,fj) + u Fi{x,/j) = 
 
 ^= -uF+u'^F-'-n''F^+ . . . 
 which series is always an entire function of u. 
 
 3. Turning to equation (2'), let us replace 
 F by uF as in the previous case. We shall 
 then have 
 
32 THEORY OF PERMUTABLE FUNCTIONS 
 
 ^ + uF + r(F$ = 0, 
 and as a consequence of the last theorem of 
 the first lecture, we have that ^ can be ex- 
 pressed as the quotient of two entire functions 
 of u. 
 
 4. As soon as we have stated the fun- 
 damental problems in this form, it is easy to 
 see that they are only special cases of other 
 classes of problems of a much more general 
 nature. 
 
 Indeed, let us consider any analytic func- 
 tion F{z-i, Z2, Zs, .... s„) whatsoever and write 
 the equation 
 
 (.3) F{,„z,,^„....z„)=0. 
 Furthermore, let us suppose that this equation 
 
 is satisfied hy Zi = Z2 = ^s =■•■■ — ^n = 0> and 
 let us regard z„ as a function dependent upon 
 2i» ^2, Z3, .... Zn-\' If the point Si = So — S3 — 
 .... = Zn = is not a critical point, we may de- 
 velop 3„ as a power series in Si, S2, • • • • ^n-i and 
 the expansion will be convergent within some 
 region. We shall thus have 
 
 (4) ^„ = S 2... S^, ■...,;__.-/.... 4-1', 
 
 -^00. . .0 ^^ '-' • 
 
LECTURE II 33 
 
 Now suppose we replace Zi, Z2, . . . . s« in equa- 
 tion (3) by the permutable functions fi, f^, . . 
 .... / „ respectively and regard the operations 
 as compositions of the first type. Then, in 
 terms of our notation, we shall have 
 
 ^(/, A . . . fn) = . 
 
 The equation which we have just found will 
 no longer be algebraic or transcendental but 
 will be an integral equation, since the oper- 
 ation of composition is an operation of integra- 
 tion. Nor will the equation in general be 
 linear as was equation ( 2 ) , but of any degree 
 whatsoever. Nevertheless, if we regard /„ as 
 the unknown function, we shall be able to find 
 its solution by the same process which we used 
 in solving equation (3). Indeed, it is only 
 necessary to replace Si, Z2, %, .... s„ in the 
 series (4) by fi, fz, fz, • • • • /„ respectively and 
 to treat the operations as operations of com- 
 position. In this manner, we find 
 
 (5) /,. = 2 2...2.1,,...,„_,/,'. ...ylV. 
 
 An interesting fact to be noticed is that 
 whereas the expansion (4) is in general con- 
 
34 THEORY OF PERMUIABLE FUNCTIONS 
 
 vergent over a limited region only, the solution 
 (5) is valid for all values of fi, f2, .... fn-\- 
 Evidently problems of integration are of a 
 more complicated nature than algebraic or 
 transcendental problems, yet we have the sur- 
 prising and interesting result that the solu- 
 tions of the former are much more simple in 
 the sense that the regions over which they are 
 valid is infinite. 
 
 We may also replace Zi, Z2, -3, .... s„_i 
 in equation (3) by ^1/, S2/2, s„_i/„_i res- 
 pectively and write the equation 
 
 Then the solution will be 
 
 Jn ^ ' ' ' ^ ^ i, i , ~1 • • • -^n-l J\ • • •/n—\ 
 
 1 • • • ?! — I 
 
 which is of the form 
 
 /»(.e-i,...2'„_i|.r, y) . 
 The series will be an entire function of Zu ^2, 
 Z3, .... s„_i , and with respect to tV and y, 
 f n {^1, 22;, ^z,- • ' .^n-i \ ^^y) will be permutable 
 with the functions /,, /2, /,i, .... /„_i . 
 
 5. AVe might also start with a system of 
 equations, as 
 
LECTURE II 
 
 35 
 
 (6) i 
 
 
 which are satisfied when ;:;i = Ui = Zo — U2 — 
 . . . . = z,, = 0. Now let us suppose that we 
 can define 
 
 f %=s 2... 2^; '..,,._ .-/■..•<» 
 {') i 
 
 Im, = XS...S.4'« , .?,''...^'.. 
 
 as imj)Hcit functions of Zi, Z2, S3, . • • • z,, which 
 have no critical point at Zi = Z2 — . . . . Zn =0. 
 Then if we write the integral equations 
 
 * * * * . 
 
 F^ (*1 /i, . . . *„ /„, <^i, . . . ^p) = 
 
 where fi, f2, fz, • • • • fn are permutable func- 
 tions, the solution of the system will be 
 
 fh = ^ ■< J (1) ^ h ~ 'n fn f\i 
 
 <^,, = S...S^lf 
 
 n fti f'n 
 
 n J \ • ••J n 
 
36 THEORY OF PERMUTABLE FUNCTIONS 
 
 and the functions thus obtained will be entire 
 functions of Zi, z^, Zz, .... s„ for all values of 
 /i> fi, fz, ■ ■ fw Moreover, these solutions 
 will be permutable with the given functions. 
 All of the equations which we have been con- 
 sidering involve only integrals for which the 
 limits of integration are x and y; that is to 
 say, they are equations with variable limits. 
 Let us see what the situation is when the limits 
 are constant. Returning to the set of equa- 
 tions (6), we shall suppose that the solutions 
 (7) are quotients of entire f mictions. Then 
 let us examine the integral equations 
 
 -^p(^l flj-" -nfn, ^l,---<t>p) = ^ 
 
 where /i, fo, fz, • - - - f n are permutable func- 
 tions of the second type. In these equations 
 the limits of integration are constant, since we 
 are concerned with composition of the second 
 type. 
 
LECTURE II 
 
 37 
 
 If now we put 
 
 
 "1 
 
 
 y.'^ 
 
 0p=2: 
 
 
 
 
 the functions thus obtained 
 
 1) satisfy the preceding system of equa- 
 tions, 
 
 2) are quotients of entire functions, and 
 
 3) have permutabihty of the second type 
 with the original functions fi,f2, • • • • /n • 
 
 Thus we see that hnear equations are only 
 a very special type of integral equations and 
 that we can pass from their study to that of a 
 more general class. 
 
 6. Let us prove certain important proper- 
 ties about functions which may be found by 
 a process like the one above outlined. More 
 precisely, let us show what certain algebraic 
 properties become when we pass from multi- 
 plication to composition. We shall begin by 
 giving an example: 
 
 We consider the exponential function 
 
38 THEORY OF PERMUTABLE FUNCTIONS 
 
 Z^ z ^ 
 
 Associated with it is an addition theorem 
 
 g(2 + 2l) = qZ g2l ^ 
 
 Suppose we put 
 
 AVe then have 
 
 (8) /(.- + ^i) =/(.a')/(.=-i) +/(.0 +/(-'.) • 
 Keeping the above in mind, let us write the 
 function 
 
 -.2 r-2. .,3 r3 
 
 We can see at once what the relation (8) 
 becomes. Indeed, we have only to replace 
 multiplication by comj^osition. We shall 
 therefore have 
 
 V(z + ,, I X, y) = V{, I :v, ij) + V (.ci I X, //) 
 
 + V{z\x,y) V[s^\x,y) , 
 that is 
 
 V{, + z; 1 X, y) = V (.. I X, y) + V (.z, \ x, y) 
 
 + f V(z\x, ^) V{,,\^, y) d^ . 
 
 X 
 
 In other words, the theorem of algebraic ad- 
 
LECTURE II 39 
 
 dition for the exponential function becomes for 
 this new fmiction a theorem of integral addi- 
 tion as we have called it.* 
 
 7. To go from the particular case to the 
 general involves no difficulty. Consequently, 
 we may state the theorem: To every theorem 
 of algebraic addition, there corresponds a the- 
 orem of integral addition. 
 
 Thus, for example, if we consider elliptic 
 functions, we can pass from these to entire 
 functions by the process of § 11 of the 
 preceding lecture. To the addition theorems 
 for elliptic functions, there correspond new 
 addition theorems for new functions. In a like 
 manner, let us consider the <j function of 
 Weierstrass. Suppose we examine for a mo- 
 ment the expansion of this function and replace 
 u in the expression by uf(iT, y) where the 
 powers of / represent operations of composi- 
 tion. The three-term equation for cr leads us 
 to a three-term equation for the new function 
 
 * Evans has studied in a systematic manner an Algebra of 
 permutable functions. (Memorie Lincei, S.V. Vol. VIII, 
 1911; also Rendiconti del Circolo di Palermo, Vol. XXXIV, 
 1912.) 
 
40 THEORY OF PERMUTABLE FUNCTIONS 
 
 which is of the integral type since we have 
 replaced products by compositions. 
 
 8. What we have said about compositions 
 of the first type may be repeated for composi- 
 tions of the second. Returning once more to 
 the example involving the exponential func- 
 tion, let us put 
 
 •* ** 
 
 jf(.~|.r,^)=./+Yr+ 3r + ■•• 
 
 This function is also an entire function and 
 we shall have 
 
 + W{z\cc,i/) W{,,\x,y), 
 or in other words, 
 
 a 
 
 It is hardly necessary to prove that the above 
 is true in the general case. 
 
 9. A similar theorem may also be stated 
 for the case of more than one variable; hence, 
 all the theorems about Abelian functions may 
 be carried over to the domain of integration by 
 a process like the one which we have indicated. 
 
LECTURE II 41 
 
 10. Let US now return to linear integral 
 equations. As indicated, equations (1) and 
 (1') are of the second kind. Those of the first 
 kind of the Volterra and Fredliolm types re- 
 spectively may be written 
 
 (9) f f(^^)n^,.^)dx = <!>{?/), 
 
 ^ 
 
 (90 f /(:l-) F(x,y) ch = cl>(^) . 
 ^ 
 
 Leaving out of consideration equations (9') 
 which can only be attacked by methods of a 
 different sort, let us consider equations (9). 
 The latter may be reduced to equations of the 
 second kind. For we can differentiate and 
 obtain 
 
 *^o dy ay 
 
 If F [y, y) does not vanish, we can divide 
 
 by F [y^ y) and get an equation of the second 
 
 kind. 
 
 J.i F (y, y) vanishes identically, the last 
 
 equation is still of the first kind. But if 
 
 i dF{x,y) \ 
 \ dy K=y 
 is not zero, then by a second differentiation, we 
 
42 THEORY OF PERMUTABLE FUNCTIONS 
 
 shall get an equation of the second kind, and 
 SO on. 
 
 li F{oc,y) is such that F (od,oo) > 0, we shall 
 call it a function of the first order. If F [x, x) 
 
 = and I — I JO , we shall call it a function of 
 
 the second order, and so on. Hence, if the 
 order of the function F [x, y) in equation (9) 
 is determinate, the equation can always be re- 
 duced to one of the second kind by a finite 
 number of differentiations, and hence can be 
 solved by the method which we have indicated. 
 
 But the order oi F (x, y) may not be deter- 
 minate. A case in point is where F [x,x) is in 
 general different from zero but vanishes for 
 certain values of x. Then the nature of the 
 problem changes, and to solve it, new methods 
 must be used. To develope these would lead 
 us too far afield. The solution of this question 
 has been the goal of numerous enquiries. We 
 were the first to take up the matter and since 
 then Lalesco and others have studied it.* 
 
 Instead of considering equation (9) which 
 
 *See: Lalesco, Introduction a la thforie des equations int^- 
 grales. Paris: Hermann, 1912. Troisifeme partie I. 
 
LECTURE II 43 
 
 is of the first kind, we may consider the 
 equation 
 
 where we can regard F and i// as the known 
 functions and <l> as unknown. For we have 
 only to suppose that aj is a constant, when the 
 equation reduces at once to equation (9) . 
 
 If we take the equation of the first kind in 
 this form, we may also write it 
 
 that is to say, the problem is of the following 
 nature: Given a function i// which is the re- 
 sultant of the composition of F and <&, and 
 given one of the factors F of the composition, 
 to find the other factor <I>. If for the moment 
 we were to replace the operation of composi- 
 tion by that of multiplication, the problem 
 would reduce to that of finding the inverse 
 operation; that is, we are dealing with a 
 problem which is analogous to the problem of 
 division. 
 
 Now it is necessary to observe that certain 
 conditions must be satisfied if the problem is 
 
44 THEORY OF PERMUTABI.E FUNCTIONS 
 
 to have finite solutions. Tlie order of xjj must 
 be greater than the order of F by at least 
 unity. For when oc=^y, xjj vanishes to a higher 
 order than F. If F is of order 771 and \jj is of 
 order n, then <l> must be of order m-n. More- 
 over, two cases may arise according as the 
 functions F and xjj are or are not permutable 
 with one another. Clearly in the latter case, 
 ^ cannot be permutable with F, otherwise the 
 resultant of the composition of the two would 
 be permutable with either. But if F and i// 
 are permutable, will <I> be permutable with F 
 and i// ? 
 
 We shall prove that this property is actually 
 realized. In fact, we have 
 
 * * * • 
 
 F^F = xfjF, FF^ = FxIj. 
 Hence 
 
 F{^F) = F{F^), 
 
 and since this integral equation has but one 
 solution, 
 
 ^ F= F^ , 
 and the theorem is proved. 
 
 11. Furthermore, when the problem of 
 
LECTURE II 45 
 
 linear integral equations of the first kind has 
 been put in the form 
 
 / i = i|; 
 other problems suggest themselves at once. 
 Thus, if F, <i> and i// are known functions, we 
 may set the problem of determining a quantity 
 such that 
 
 (10) F X+ X^ = i/;; 
 
 or again the following problem: given the 
 known quantities .F:^, Fo, F^, F^, and i// to 
 calculate a quantity $ such that 
 
 (11) >, i + i >3 + #3 i i^4 = i/; . 
 
 The above are new equations which up to the 
 present have never been studied and with which 
 we shall now concern ourselves. 
 
 First, let us consider equation (10) which 
 we can write 
 
 Let us put 
 
 X 
 
 -fx{z;^)^{^,y)d^. 
 
46 THEORY OF PERMUTABLE FUNCTIONS 
 
 Then we shall have 
 
 g^ = ~F{x, X) X{x, II) +/^ F,{x. I) X(^,^)^£, 
 
 ^= -^{U^U) X{x,?/) - J A%r,|) ^,(^,y) <Ii+'^ , 
 where we have put 
 
 dF a^ 
 
 ' dx ^ ^ d Jl 
 
 From the first equation, we derive 
 and from the second 
 
 + jy{L^/)^-^^d^+ii{x,j/). 
 
 where / and ^ are two known functions which 
 one can obtain from F and <I> and where 
 
 is also a known function. Then by subtracting 
 tlie second equation from the first, we have at 
 once 
 
LECTURE II 4T 
 
 dS{x,y) _ 1 dS{x ,?/) _ jjf^^^ ^^^ 
 
 ^il/^U) ^// F{x,x) dx 
 
 -\- J nx,^)-^Y d^ 
 
 -J ^.W'-^^^;^ ^' 
 
 and integration by parts gives 
 
 Thus we are led to the following result : To 
 solve the integral equation (10) we must solve 
 the problem which presents itself in the shape 
 of the last equation. This problem is nothing 
 more than the integration of an integro-diff er- 
 ential equation. Indeed, equation (12) is both 
 of the type of an integral equation and of a 
 diiferential equation. 
 
 The above problem admits of a solution, 
 but we shall not go into details of the solu- 
 tion. The interesting point to notice is that 
 integro-differential equations arise in a great 
 
48 THEORY OF PERMUTABLE FUNCTIONS 
 
 variety of problems. We have examined these 
 equations in a number of forms and have made 
 a particular study of the integro-differential 
 equations of the second order and of the ellip- 
 tic or hyperbolic types which arise in connec- 
 tion with certain problems of mathematical 
 physics.* 
 
 The problem we were considering is of a 
 different type. It is of the first order, and 
 since two dependent variables appear, it cor- 
 responds to problems involving partial deriva- 
 tives. The case of equation (11) may be 
 handled in a similar manner. 
 
 12. We wish to demonstrate certain inter- 
 esting results which are closely connected 
 with the problems we have been discussing. 
 Let us go back to equation ( 9 ) . In certain 
 cases, this equation has a finite number of solu- 
 tions, while in others the number of solutions 
 is infinite and the solutions involve an arbitrary 
 function. 
 
 To see this, we need only to consider the 
 equation 
 
 * I.e^'ons sur les fonctions de lignes. Paris: Gauthier_Villars. 
 1913. 
 
LECTURE II 49 
 
 and to determine under what conditions x ~ ^ 
 is the only solution and under what conditions 
 solutions other than exist. 
 
 To simplify matters, we shall assume that 
 the functions F and ^ are of the first order 
 and shall determine under what circimistances 
 the equation has a solution of the first order. 
 
 Suppose we write oin- equation in the form 
 
 (B) flF{:r,^)x{^,f/)d$-{-f'x{.r,^) ^(^,y) d^ = 0. 
 
 Then by differentiation with respect to y, we 
 have 
 
 and when oj = y, 
 
 which gives us a necessary condition. 
 
 Moreover, by suitable transformations of a 
 simple sort, we are always led to the case where 
 
 (12) F{:x,x)=-^{x,x) = l, 
 
 (12') F,{x,ai)=F,{x,x) = ^y{x,x) = ^,lx,x) = ^ , 
 
 where the subscripts 1 and 2 denote partial 
 
50 THEORY OF PERMUTABLE FUNCTIONS 
 
 differentiation with respect to x and y res- 
 pectively; that is 
 
 For we can first write 
 
 ^ = f(^i) , y = /(.yi) , 
 
 X'i^i,//i)= ±J/'(-n)/'(//i)x(-^.^). 
 
 Then if we take 
 
 _ ±1 
 •^'(•"-"^^ ~ F(^) ' 
 we clearly see that the equation (B) becomes 
 
 where 
 
 Hence we can suppose at the outset that con- 
 ditions (12) are satisfied. 
 
 The above having been established, equation 
 
LECTURE II 51 
 
 (B) may be written 
 
 jy{,^ x{x, f) m ^{^, u) ^f ^^^-=0 
 
 m 
 m 
 
 If we put 
 
 we shall have 
 
 But we can make use of the arbitrariness of a 
 and ^ to choose 
 
 F\{x, X) = F'lx, X) = ^'i(.i-, x) = ^',{x, x) = , 
 
 which shows that we can always assume that 
 condition (12') is satisfied. 
 Now let us write 
 
 We shall have 
 
52 THEORY OF PERMUTABLE FUNCTIONS 
 
 whence we derive 
 
 where 
 
 /•i(^,^) = -^i(.r„//) + /f(.^V/) + ^i%r,,y) + . . . , 
 
 and therefore (see Lecture II, § 1) 
 fi{x, x) = <f>lx, x) = . 
 Hence, integrating by parts, we have 
 
 where 
 
 and therefore 
 
 ^-g + /; [X(.r, f ) ^ily) ->l>{.v,$) K^.i/)'] <'f =0, 
 
LECTURE II 53 
 
 This integro-differential equation may be in- 
 tegrated. 
 If we write 
 
 G{x, y) =fj [X(.t-, I) i/,(|, y) - xp{:v, f) [i{^, ^)] d^ , 
 
 we have 
 
 (16) i// (.r, /j) = div) - f" G{^ + v,v-^) d^ 
 where d is an arbitrary function and 
 
 The sohition of the equation (16) is obtained 
 by the method of successive approximations. 
 
 ApjDhcations of the above will be brought 
 out in the next lecture. 
 
LECTURE III 
 
LECTURE III 
 
 1. We shall begin with some applications 
 of the work developed in the last lecture. 
 
 We have solved the problem of finding the 
 function x ('^'^ //) which satisfies the equation 
 
 on the hypothesis that F and <J>' are of the first 
 order. Now suppose we put 
 
 Then the condition 
 
 <^{x, X) + F{x, X) = 
 
 is clearl}^ satisfied, and hence we shall be able 
 to calculate all of the functions xi^^V) which 
 satisfy the relation 
 
 in other words, all of the functions which 
 have permutability of type one with a given 
 function. However, in the last lecture (§ 12) 
 this problem was solved only in the special case 
 where the given function is of the first order. 
 If the function is of higher order, the method 
 breaks down. 
 
58 THEORY OF PERMUTABLE FUNCTIONS 
 
 We have seen that the problem may be re- 
 duced to the solution of an integro -differential 
 equation of the first order. If the given func- 
 tion is of the second order, the integro-differ- 
 ential equation which we must solve is of the 
 second order and admits of a solution. An 
 arbitrar}^ function always enters in. 
 
 As we increase the order of the given func- 
 tion, the problem becomes more and more com- 
 plicated, hence we shall not go into details on 
 this question as we should be led too far afield. 
 In the general case where the functions are 
 analytic the question has been answered by 
 M. Peres.* 
 
 2. We wish to present some of the prop- 
 erties of permutable fimctions. The very 
 method which enables us to calculate all of the 
 functions that are permutable with a given 
 function also leads us to the result that if 
 F and <I> are permutable and if F is of the 
 first order, then we must have 
 
 ^{x, x) . 
 
 ^ ' — - = const. 
 
 F{x, x) 
 
 We shall give a rigorous proof of this fact. 
 
 * Rendiconti dci Lincei. 1913-14. 
 
LECTURE III 59 
 
 We write 
 
 Differentiating with respect to y, we have 
 
 F{x,y) ^(y,y)+f'F{x, I) 4>o(|,,y) d^ 
 
 = ^(x,//) FO/,//) +f'nh I) F,{^,//) d^ , 
 
 and differentiating this last expression with 
 respect to a:, 
 
 F^{x, y) ^{y, y) + F{x, x) <^^{x, y) 
 
 - <I>(:l-, .T) F,(x, y) + /J<&i(:*', I) F.i^, y) d^ . 
 Suppose we put <r = y. We shall then have 
 Fiiy, y) ^{y, y) - F{y, y) 0,(,y, y) 
 
 = ^i{y, y) F{y, y) - <^{y, y) F.iy, y) , 
 that is 
 lF,{y,y)-^ F.iy,y)]<^(y,y) = 
 
 F{y,y)[^^y,y) ^^^iy^y)] . 
 Moreover, if we put 
 
 F{?/, y) = Ay) , K!/, u)=<^ {y) , 
 
 we have 
 
60 THEORY OF PERMUTABLE FUNCTIONS 
 
 and consequently 
 
 whence the theorem. 
 
 3. It is a simple matter to Und the expan- 
 sion of any function xjj which is permutable 
 with a function F of the first order. 
 
 For by the last theorem 
 
 \lf{x, x) _ 
 F{x,x)~^^ 
 
 where Ci is a constant. The expression 
 
 y\f[x,t/) — c-i F{:c,/j) 
 
 will be of higher order than F and permut- 
 able with F . 
 
 Now by one of the theorems which we proved 
 in the last lecture, w^e may write 
 
 where ^i will be permutable with F . Then 
 there will be a constant c-z such that 
 
 and consequently we shall have 
 
 >\, = c,F + c,F~ + ... 
 If this process can be carried on indefinitelj% 
 
LECTURE III 61 
 
 we shall have under certain conditions an ex- 
 
 * 
 
 pansion of i// in terms of F, F'^, .... 
 
 4. We shall give a short survey of the 
 results which can be obtained by the intro- 
 duction of a new symbol. If we put 
 
 we can write 
 
 F= 6-^ ^, ^ = ^F-\ 
 where F ^ and ^'^ are merely symbols which do 
 not represent functions but which may be 
 treated as if thej^ did. If the functions are 
 permutable, we can write 
 
 F = ^i-^ = 6-^^, 
 and if 
 
 Fie = x, 
 
 hence the symbols <I>'^ and F^ are themselves 
 permutable. 
 
 Let us assume that we have permutability. 
 We wish to determine 
 
 ^-1 + / 
 
 j-i 
 
62 THEORY OF PERMUTABLE FUNCTIONS 
 
 In other words, let 
 
 We shall then have 
 
 and owing to the property of permutability, 
 
 >i(0i + 02) = {F + ^)rjf, 
 
 whence 
 
 01 + 00 = {F<I>)-'{F^^)^ 
 
 = {F + ^) {F^)-^^ , 
 and we may write 
 
 ^_l _^ j,_i _ (/^^_<|,) (i^i)-! ^ 
 
 that is, if/z^ rule for the sum of tivo fractions 
 may he applied. 
 
 Tlius we see that we can develop as it were 
 an arithmetic for the symbol F ^ quite anal- 
 ogous to the theory of fractions. 
 
 5. We have seen (§ 1) that if ^J* and \\t 
 are of the first type and if 
 
 ^(2-, .r) =- T//(.r, .?;), 
 then a function i/; {a\ if) may always be found 
 such that 
 
LECTURE III 63 
 
 Hence we can write 
 
 (1) x = i-'x^ = ^xr', 
 
 ^ = x^x~' ■ 
 
 And by solving the equation 
 
 we shall have that 
 
 (2) * = y-.xf, 
 
 Therefore the two functions ^ and r/» ' can 
 always be obtained the one from the other by 
 a transformation through the functions x or x'- 
 
 In particular, if 
 
 xjj{x;.f) = 1, 
 
 we shall always be able to find 
 
 (3) X-iix-' = l- 
 
 The relations (1) and (2) may be obtained 
 even if ^ and xjj are permutable. In this case, 
 X and x' do not belong to the group of func- 
 tions which are permutable with the given 
 ones. In particular, equation (3) may hold 
 even if \jj is permutable with unity. 
 
 6. We shall bring these lectures on per- 
 mutable functions to a close by extending some 
 
64 THEORY OF PERMUTABLE FUNXTIOXS 
 
 of the results which were obtained in the first 
 lecture. (§ 11.) 
 
 A function which depends upon all the val- 
 ues of a certain function /(ct) between the 
 limits a and h admits of an expansion 
 
 (4) A,+j\f[.v,)F,[x,)dx, 
 
 + r ' r " ./■(•'•i) f (-^'2) -^2(-^i, •'^2) dx, (h, + . . . , 
 
 provided certain conditions are satisfied; 
 where F 2(xi, 0C2) andi^o(cri, oc-^, oc^), etc., are 
 symmetric functions. The expansion in ques- 
 tion corresjjonds to Taylor's exj^ansion (or to 
 a power series) in ordinary analysis.* 
 
 With these facts before us, let /' (.r, // | a) 
 be a set of permutable functions of type one, 
 that is of such a sort that if a be given any 
 two values ai and a.^. the two functions tliereby 
 obtained will be permutable with one another. 
 
 As an example, we give 
 ./■(•'—// 1 a) 
 which has the above properties. 
 
 * See: Legons sur les Equations integrales et intigro-dif- 
 ferentielles. Paris: Gauthier-Villars. 1913. Chap. I, § VIII. 
 Legons sur les fonctions de Ugnes. Paris: Gauthier-Villars. 
 1913. Chap. II. Lectures delivered at Clark University, 
 Worcester, Mass., 1912. Third lecture, § IV. 
 
LECTURE III 65 
 
 Now, let US write 
 
 ^ X 
 
 //(•^^li8)/(l,.y»./|=y(.r,^ja,/3). 
 
 The function j{oc, y) | a, ^S) is permutable with 
 the original ones. 
 Again let us write 
 
 and so on, and let us suppose that the series (4) 
 is convergent when \ f {x) | is less than a cer- 
 tain quantity. Then if we write the series 
 
 Jb ^, b 
 ./'( -f^// I -^'l, •^'2) ^"(-^'l v'^-2) t^-^"l (^^2 +'", 
 a -^ a 
 
 it will converge no matter what the absolute 
 value oi f{oc,y \ a) may be. 
 Moreover, let us consider the series 
 
 (5) ^{^)=/{^} + f\n.r,)F,{.r,,^)J.v, 
 
 •^ a ^ a 
 
 If the determinant of the linear integral 
 
66 THEORY OF PERMUTABLE FUNCTIONS 
 
 equation which we obtain by taking into con- 
 sideration only the first two terms does not 
 vanish, we can derive /(I) as a function of 
 <J> ( I ) from equation ( 5 ) , provided | <5» ( ? ) | 
 does not exceed a certain vakie.* 
 But let us examine the series 
 
 « 
 
 Then if ^ is known, we can derive f{x,y \ |) 
 in the form of a series which converges no 
 matter what the absolute value of ^ {x,y \ ^) 
 may be. 
 
 This is the latest theorem which we have de- 
 rived in the field of research we have been 
 developing. 
 
 * Leqons sur les Equations intdgrales et integro-diff^rentielles. 
 Paris: Gauthier-Villars. 1913. Chap. Ill, § XVI. 
 
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