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1 
 
1-0 di. 
 
 
 h’IRST LECTURE' 
 
 Solution of equations by passing 
 to the limit from Approximations.; 
 
 The problems that arise in algebra are of a finite character. 
 
 These meet their first generalization in the study of infinite series 
 in which the notion of sum is generalized from the finite to the 
 infinite case.; We meet a similar generalization in the Riemann 
 definition of integral.’ In this well*“known case an interval of 
 integration^ from to Xn j is divided into ri sub— interval s^ the 
 interval from Xj^ having a lent^th ,* 2 ^. The function value 
 
 of the integrand is taken at some point in the interval^ and 
 
 the sum formed 
 
 ^ 1,. C, n. 
 
 The limit of this sum if it exists^, for any method of subdivision 
 into n parts as n increases indefinitely^ the length of the maximum 
 'hj decreasing indefinitely with 1 ^ n, is the Riemann integral.' 
 
 Likewise in Jefinin^ a derivative we pass from a finite case 
 to an infinite case.; Usually the derivative is the inverse of the 
 integral defined above^ but it was pointed out by Dini that there 
 were functions which had derivatives but such that the derivative 
 was not inte^rable in the Riemann sense,' On the basis of his own 
 and also investigations of Sorely Lebes^ue «jave a definition of 
 intCiJral which was much 1 reader in extent^ and took care of some of 
 these exceptional cases ; 
 
 The definition of derivative is usually ^iven in the form 
 
 F( X* h)-F{ x) 
 
 This can be determined approximately by taking \ as small as possible 
 

 
 4U 
 
 ^iifilaa^o %<,i isnoiifEupa lo noiJuio2 
 S«floli£(£iixo'iaqA atot^ iiiail 
 
 ' 3 'i ‘'\ 
 
 
 H, 
 
 *5^iffx3 £ ![o 97K ut^ajda ni &8ti& ^utrl^ eojaldoiq S 
 
 :Wji 
 
 .SaiTae «-‘i(.Mni to ,iuJ8 ni noxias ileisnss Xanii nifux,,, 
 
 9.-IJ oJ sXini^ sdj „otl tosilaianss si lua }o notion erfj iioHw n 
 
 ■■•j ■■ ' li ^ 
 
 nna^nsifl srIJ ni noiissiXe-isnss isliaia a Jssn sW i.saat aiinilni 
 
 RV, 
 
 io Xavisini na asao n»onH-XX,» airii nl ;.Xeis,s.ini io poilinilal 
 arii ..sla.naini-dua „ oini iabivit^si oi ,x »on1 .noixan^ain) 
 »tlav noixonai aPT .jP rfx.nnj , jx oX ,.jx noiT Xa.neXa^ 
 
 fcns ,i3 Jaanaini arix „i xnioo aaoe Xa naSaX si tn.n^^fnt ,rtx Ir 
 
 b^OIlO^ nog 8(iy 
 
 
 =J 
 
 
 
 noiaivifcdua tortXaa yna no! ,.alaixa xi tl Ln ,i,1x i, XiaixVl 
 
 »aaixax. arfx lo rixpnaX arfx .vXaXiniiafcni easaaaoni n sa aXnac x oJnX 
 
 'V 
 
 
 ^ -.flansaxni nnaaaiR anx ^i V i ylaxlniXatni Jnisaanaafa 
 
 asaa aXini.i a »oai aaaq a« ayfXavlnab *■ JninilaX ni aeiwaJtU , 
 
 ..Xi 50 asnavni aPx ai avixavinat a,.X yllanaO ,aaaa aXinlXxfi na oX 
 oxanx XartX inxQ yp x„o iaXnioq aaw ii' XoX- ..avoXa ianilat Xansaxni 
 _oviXaviaai aXX Xa.iX Pana Xnd saviXavinai fad daid. anoiXonni 
 
 i ’■ 
 
 iild 
 
 -o Sid 10 sisad.adX „0 ;.aanas nnaoaiXi adx ni aXdaa.axni x\n' ..a 
 50 noixinilat a avas anvsadaj .,lano8 1o snoiXa sixsayn i .^'xa .b„^ 
 ■io a.os 50 anaa dooX ina .,x„,xxa ni nataon - dono san doidw lanyaLi 
 
 lsnoiSo 90 X 3 0 B 9 fiJ 
 
 _,x.dJX adX nl navxi ylXsuay ai aylXa»inal >o noiX-iniXai edl 
 ' tx ♦x i'** 
 
 I i&Il. " Vd Vlalaaixodooa iani.-.axoi ,d n«, 
 
though not «ctuaUy reaching the llmitini case of '^-o,, Such approx- 
 imations are quite useful in aopZiei mathematic a,! and for many such 
 purposes satisfactory^ 
 
 If now we consider the simple differential equation 
 
 dy/dx • f(xyjf with the usual conditions on the functional Cauchy 
 showed that an lotetjtral always exi'sts.j The method generally used 
 
 for the mechanical integration of this equation is an apDroxiiDation!,i 
 which pushsl to the limiting ^ase shows the eadstence of the function 
 called the integrsl!.J This can be done graphically by strating from 
 a given point acoi,i yo-r^ calculating by the equation the slope of an 
 integral curve at this point.,1 then continuing in the corresponding 
 direction for a short distance tc which point a new slope 
 
 is calculated*] This process repeated for short enough intervals 
 will lead to a close approximation to the integral curve.! In symbols 
 we may Indicate this process as followsi, 
 
 W(- ^ 
 
 •«i-4 
 
 from which we have a set of equations of the form 
 
 y{xi)-y(xi^lhf(xi^i,yi^l)(xi- 
 
 If the successive values are substituted in the equations 
 we arrive finally at the value of y(xn) in terms of a sum of 
 products of the form 
 
 *e may extend this method to aoply also to the oase of 
 
 systems of equations.) For instance let 
 
 iy/dx - 
 
 dx^'iix ” •t,i(x)y * 
 
 Proceeding to exoress the approximate results for n different values 
 of *, and multiolyini out the results we have 
 
 yn® (l + .’Zn 
 

 yThe passage f ro’n Zr._i to Un j Zn is by rceans nf the substitution 
 
 **12 
 
 " ■ ... 
 
 # 
 
 ,"'To pass then frofu \io Zq to jt,>, Zn woul .1 require a succession of these 
 substitutions 
 
 • 3 2 3 1 . 
 
 cr 
 
 -n 
 
 ^ We nay write 3 n in the forno 1 +’:^^ where is the matrix of the u's 
 
 ! alone^ and the product of the succession of substitutions takes the 
 j f orm 
 
 . 1 ■*■ t . vy ( x • M ) + m '2 ^ yiy ( ;c • j f X y j * • • • 
 
 iHhe summation bein^ from 1 to n. If we pas:s to the limit, we ar^'ive 
 
 at inte->'rals of a substitution 
 
 1 + f ( x>) ( ) dx ^ r r t f Jc I ) f X 2 ) ( ) ti X idx 2 + * • • 
 
 [See Volterra: Leccns sur les functions des li.>'ne 3 ^ 00 36 — 42 ] 
 
 f' Continuing our develoonrent^ let us notice next the 
 
 (■^equation d^^^dx = f(x)u(x), of whic'. we may write the integral as an 
 lithe solution of an inte^'^al equation.^ namely 
 1 h(x) = + c. 
 
 |lr a more general fo'^m we would have to consider as linear integral 
 ■equations the forms 
 
 ^ix) = f{x)-( x)d^ the Volterra equation of second kind^ 
 
 x)= f Q^(t)F(hjx) di. the Volterra equation of first Kind^ 
 
 x) = f(x)~\^ 0 ^>{^-)^'i^,x)dF the Preihclm equation of the seccn 1 kind, 
 
 L '^{x)=\f (i^^f{t)F(^.x) dh the Fredholm equation of the first kind. 
 
 ! To these we may aooly the same metho ds of constructiru' a solution by 
 
 j the methods of aoproximation under discussion^ viz. those which 
 i consider the values of the quantities at n ooinl, in the region under 
 u consideration^ passing ttience to the region by lettin.? n becoiiie 
 ^'infinite. An exaiole anicnu tne eanliest integral equations is 
 that due to a problem of Abells. The pro b lei; is this. deter vine 
 that curve in a vertical plane iowti which a xovin.? 00 dy would pass 
 iin such a way that it would reach the lowest point in a time ^iven 
 
 
by th« function whore '*1 is the initial heij^htui the initial 
 
 velocity bein? zero.J If wOh) i's constant for all heiafhts the 
 carve is the tautochrone,\ The integral equation which has to be 
 solved is 
 
 / x) " To ^ 
 
 ✓ ( 
 
 Solution;, uix)*’ ^ 
 
 A dx /Kx-hJ 
 
 y‘u(x) is the equation of the curve. | 'ISee Bocherij.t Linear Intesfral 
 Equationstji Volterral,' L'econs sur les equations inte|rale9.j] 
 
 A generalization of this is the eqaati<}n 
 
 f(x) * f xj { dB, where 0<a<l,j 
 The solution of this is 
 
 ^ fl^ Abel"s formula.] 
 
 n dx vx- “ 
 
 These equations treated by the eethod naentionei oay be found in 
 Volterra:. Lecona sur les Equations inte^rales pp 34 etc.] 
 
SECOND LECTURE 
 
 We return ^ow to the equation of the second kind 
 The solution of this equation is found to be 
 
 '^ix) * rraf;+/*o^s(Ex;r(a)de 
 
 where the function S is sfiven by 
 
 %{x]i) « -f{xy)4*ixy)^h{xM)*~^^^r^ 
 
 In these equations f is bounded and inferrable. ; The series for 3 
 is uniformly converrent.; The meaninr of the ^ over the F is as 
 
 follows: We define for two functions (waiving for the oresent their 
 
 character) 
 
 which is called oomvoaition of the first kind. In comoositlob of 
 the second kind the limits of integration are fixed j This ooer^tion 
 is associative^' but generally it is not commutati'^e ' When the ' 
 functions and P, permit of an interchanre without chanrinr the 
 resultinr function they are said to be oermutahle. [See Colterra^,- 
 Fonctions des lirnesy Chapter IX.;] If there is riven a system of 
 oermutable functions^- their compositions are also oermutable with 
 the oririnal system and with one another.' Also the system arrived 
 at by addinr them or by addinr constant multiples of them is also 
 permutable. with the others.; The rreat advantare of oermutable 
 functions is thefact that the ordinary rules for handlinr alrebraic 
 rational expressions apply to them^ and with proper interpretation 
 also the irrational processes. 
 
 As an instance, all the functions oermutable with T,- 
 unity,' are of the form F(x~y) or F(y-x), 
 
 The fundamental theorem which bears upon expansions in 
 
 compositions is this.; Let 
 
 OC OC <30 
 
 oi 10 i oifi ^ i 1 1 2* I* 
 
 
 
 a 
 
 I* t* t Zfi 
 
 be a serie** of powers of the complex variables Zi Zn which is 
 convergent for values of \z^ ^ ffi Tf'lznl however small 
 
 i?i /?, 2 ,' -rr! En provided they are actual finite values,- then if we 
 replace the variables z by functions Pi^' , and place a s^ar 
 
— 6 - 
 
 ‘ov«r each F so that the oocration changes from multiplication of 
 variables to compo si tioii of functions, the resulting expansion is 
 uniformly conversfent for all values of the variables x,y and for 
 any functions F which are bounded.! Volterra:. Fonc i li^nes p 140 .f 
 
 This theorem included Borel's expansion theorem as a special fonj.; 
 p.| 160 loc cit.j 
 
 Returning now to the equation of the second kind 
 
 r(x) = ro^^(i)F(tx)di 
 
 ie arrive at an equation of the second kind,* and may then solve, 
 frequently by dif fer- ntl atini both sides as to x, ^ivin^ 
 
 f'(x) * x)F( x,x)-^f 
 
 It is evident however that if F(x,x) * 0 we are no better off than 
 before:,! as the non—homoiieneous term drops out.j When this value is 
 such that 
 
 F(xtx) ^ 0 F is of order 1.; 
 
 It is clear that we have F(x,y) * ( x, y) where d(xyj is not 
 
 divijiib’le by (y^x), does not vanish for in which case we say 
 
 F is of order w. In case ® is an integer it is evident that m-fold 
 
 differentiation of F will lead to anequqtion of the type desired,! 
 so that we can resolve the problem.! There are left to consider 
 
 other types of nucleus of this form', particularly those that have 
 algebraic 6r logarithmic singularities for y"x, ISee Lalesco, Jour.] 
 des Math. I 1908 op 125-2021 .j In case F(x,x) vanishes at a set of 
 point X which are not a continuum the order becomes indefinite.' 
 
 Lalesco has considered such forms,! 
 
 In the case f(x) * fo F{tx) ^dB. 0<a<l 
 
 «e multiply both piles by anl intelrate fron 0 to p. 
 
 The left sile is a Known function say ^(xJ. The rilht side/ by 
 interchanging the integrals lakes the torn 
 
 f(lx)(y-xJ'‘-Ux-t)-'' dx 
 
 In deter.ininl the new limits we use a theore* lue to Oirichlet 
 In fact the inteiration is easily seen to take olace over a 
 
 iri an^l e 
 
- 7 - 
 
 havirii? as sides the u axis^ the bisector of the an^jle between thh' i? 
 xaxisandthej;/axis^anialine^/=a. 
 
 The expression F ( e,x) ( y-x) ^ evidently has the same 
 
 order as F, We set the last integral f - § {B,y) an expression of 
 the first order^ and then have our original equation reduced to one 
 of the first kind which is manageable/ [See Lalesco's paper above/J 
 We come now to the case with a logarithmic singularity: 
 
 f(x) = To '^(x) [\oi,{x-B.)*0\dl where C is ta^en actually Euler's 
 
 const = 0 .•57721--r-, 
 
 1- The method of recudtion of this equation actually presented is not 
 ^ the one used in the original study, »and aoioea^s therefore to be 
 h highly artificial.' We staBt with the integral 
 
 i: ! - 
 
 1 r(a)r(e) ^ r(u+e) 
 
 , This is found easily by setting t~x~(ij-x)z in the form 
 
 
 rart_ 
 
 ru^t) 
 
 The left side of the equation written may be split into 
 two parts, one containing u the other fc* only. We now differentiate 
 as to « and integrate as to b from B to */ The result on the ri^ht is 
 
 -iy -x) 
 
 The result on the left consists of the two partstreated separately,, 
 I the one containing u the other b/ The differentiation as to u 
 reduces that part to 
 
 '(Fu)'" 
 
 The other part we write /\ 3 
 
 We now take a=l 3=1, and we finally have the formula 
 
 f ■*'C) /\( y, di = -(y-i-x) 
 
 Now if we return to the integral equationn under consideration we have 
 
- 8 - 
 
 Chani^in^ the variables we have 
 
 (E-x)-*-C]/\(j/^)'i^.) = '-{y-x) ^ 
 
 From this 
 
 r o^f(x)/\( yx)dx =r 0^ /\( yx) dxf o^'viO [loi?(x-£.) +Ml dE, 
 
 
 ~f o'^ 4 '(^)d^r yx) f,lo^(jc-£,) +CJ dx 
 
 .The left side therefore reduces to -ro (^) ( its derivative to 
 y is ‘-f dtj> and second derivative as to y is •~^f(y). 
 
 IS 
 
 in 
 
 !'• 
 
 Nr 
 
 '•j 
 
 TEIUD LECTURE 
 
 t, 
 
 The process utilized to solve the equation last 
 considered is an example of a very general process by which an integral 
 .equation with a ?iven nucle.us may be reduced to anoL.{\er with a simpler 
 nucleus.; In fact if we have the equation 
 
 r(xJ = fo^ si,{OF(lx)dl 
 
 and multiply on both sides by a ^iven function ^ (xy) then integrate 
 
 as to X, we have 
 
 (xy)f(x)dx=fo y§(xy)dxf o^^>{OF{tx) dt 
 = f o^^fiOdtf ^^F(E.x)^( xyJdx 
 = /’o^^y(e) ^{ly)di 
 
 We may set the result of the composition Ff=^(xy), thus arriving at 
 a new equation with a new nucleus.' An example is the previous case 
 where the function equation with nucleus [ 1 oi ( jc-£. ) 
 
 into one with nucleus -(][/-£.).' We now purpose to find a function ^ 
 which will convert the nucleus \oi( x-i.)*0, where C is the Kuler 
 constant 
 
 We start with the formula 
 
 r 
 
 ' X 
 
 
 
 dt 
 
 (y-xj 
 
 a + b— 1 
 
 Tu re r(a*e) 
 
 The meabers of this equation are icultipliei by • the parts 
 
 on the left beini separated • We then repeat the process used before 
 of differentiation as to u and integration as to b form to . If 
 
1 
 
 t 
 
 \ 
 
9 - 
 
 T 
 
 we reoresent by P the expression 
 
 B B pf, 
 
 then we have 
 
 r(a+e) 
 
 Nowseta=l B=landwehavethedesiredresult 
 
 fx^[\oiU-x)-V'l/Tl +^]P(£f^;cia=-(.v-Jc) 
 
 lln these equations if we do not take a=l we may ccaiDlicate the nucleus 
 lito a considerable extent,; 3y further differentiation as to a and in- 
 |te?ration as to b it is possible to arrive at expressions involving 
 the square of the lo^arithiu^ and even higher powers and polynorcials 
 in the lo^.; 
 
 This leads us to inquire .what the nost general integral 
 equation may be like. One case studies by Lalesco (Equations Inte»?rales 
 p 127 et seq) i s 
 
 lj(xj = f 
 
 This is not linear^ and indeed F may be quite 4eneral Another general 
 form would be 
 
 ^>(x)=^y(x)+fo "-F i{E.x)di.*f o^'f 0 "-dS. i,di. s{e,j.x ^x) j.) yii s) ^ 
 
 We need not multiply these special cases but pass at once to the 
 consideration of the most general case that can occur.; We find necessary 
 here the general conception of function of a line, and that the most 
 ieneral integral equation takes the form 
 
 ^)(x)^ ^ [yit), x] 
 
 the riJht hand side representing a function of all the values 4iven 
 by the function yU) where ^ takes every value from 0 to 1, and a 
 
 parameter ;c, which may also take every value from 0 to 1.; This equation 
 may now be studies as an equation which is a limiting case of an 
 infinity of equations depending upon an infinity of variables.; 
 
 See Volterra, Archiv der Math/ und Phys.; (3)23(1914/5) Sitzber.; 
 
 Berlin Math Ges.; 13(1914) pp 130-150. ; 
 
 ' 0 ^ ■ 
 
 /fr 
 
 U.. 
 
-10- 
 
 ' V « 4 !*• 
 
 Let u(x) define a curve^^ then any set of numbers determined by some 
 process of determination such that for each function u(x) there is 
 one number, ‘ are the function values of a function of a line.' The 
 argument in this case of the function is the line in a certain sense, 
 or we may say is the totality of all the function values of yix) 
 for every vaOue of jjc from 0 to 1.' A simole example is the area under 
 the curve,' another is the length of the curve.; In physics we have 
 many such cases,- as for example the ootenytial at a ^iven point in 
 space due to a loop carrying an electric current is a function of 
 the loop.; 
 
 We may study such functions by our method of development 
 depending on a set of n selected points and the corresponding 
 ordinates,' so that we would have a function ^ ( Ij u Us j ' '' ■' 
 Afterwards we determine the limit of the expressions as n approaches 
 infinity.; We may also have a set of such function 
 
 ^ y xj y 2/ '.Vn ) “ z 1 
 ^ 9^y X} y 2 j * i'n ) - z 2 
 
 r • ? • r • ♦ 
 
 • f 
 
 ^ y Xf y 2 j ‘ ‘yn Zn 
 
 If we attach the values z to the points alon^ the line we may look 
 upon them as values of a function z( x) , and must then con adder that 
 in place of the index of the F we must introduce jc as a parameteif. 
 
 we have as the limitini? case 
 
 FlyiOjx] - z(x) 
 
 • • 
 
 where the left side is a function of a line depending upon a 
 . parameter x. This is the most general form of an integral equation 
 ** ( of the Fredholm type,- but this includes all the types.) 
 
 ^ In order to solve an equation of this form we expand it 
 
 || in a form similar to a Taylor Series ; In fact the exoansic -i is a 
 ft generalization of Taylor's series from the case of variables to 
 

 
> that of an infinity of variables.' We exoand the function a 
 1 14 . he in the form . ^ 'J 
 
 
 ^ z(x)=y( x) +f 0 o^f o F git\^ gx)'dCl'iE,.y+" - 
 
 It should be noticed that if we stoo with the first Integral we' have 
 
 
 the linear integral equation.* A resolvent function;bis founds and 
 we then have 
 
 yi x) = z(x) + f dti + f ^zilx^gx) citidts-^' 
 
 See Volterra Fon' tions des li^nes p.; 69 et seq.; 
 
 FOURTH LKCTURE 
 
 Inte^ ro-di f f erenti al equations * 
 
 The equations contain the unknown function not only unier the 
 integration si^n but also differentiated .partially or totally with 
 
 respect to the arguments it contains * As examples 
 
 df* dx'-^ to 
 
 which ^ives the motion of an elastic cor 1 when hereiitary effects are 
 
 considered.,- that is when the motion changes the character of the 
 elasticity progressively.,* oroduciriij an effect similar to hysteresis 
 in magnetism.’ See Volterra Rend.; Ac | Real ; lei Lincei 191^.,'/ 
 
 Fonctions des li^nes p 97.; 
 
 Another example from physics is 
 
 t 
 
 £.kiLl = n(t)*fo\'('^)F('it)d^ 
 dt'^ 
 
 to which the solution of the precedinu is reduced.; 
 
 From analysis we easily find examples.; We shall consider 
 
 here only t that one arising in the solution-of the problem of 
 finding all the functions oermutable with a ^iven function.' We 
 recall the definition of permutability 
 
 (xOfO-uidf- 
 
 ks an intpolactorv Raamole which will suiiest the general aethol 
 of attaol^ let as take that is we oropose to fini the functions 
 
 oermutablewith v , \ ^ 
 
 ^(xy) = ^(xOdt 
 
 I If we differentiate as to k we have at once 
 
 if Differentiatini as to y we have also 3f 31/ i ^ »U 
 
 these we eliminate the funotioh and have a partial 'i 
 

 - 12 - 
 
 for whence 
 
 4 / * F(y-x) " 
 
 lere F is arbitrary but differentiable.; These functions f urt hertnox e< 
 ia1*e a>Jl oermutable with one another,' and in this sense foriE a ^rouo. 
 
 his is called for physical reasons the s^roup of closed cycle, 
 
 (Volterra Fonctions des li^nes Chap.'VIl),; The auestions as to whether 
 it is true that the iroup of functions peirmutab le with 3 ^iven function 
 are therefore permutable with one another has been answered affirma- 
 tively by Vessio’t^ C.;R.; 1912 p 682.; In this i^eneral case we make 
 use of a transformation upon the whole sis follows,' Let F(xy) 
 
 be one of the 5 roup.; Transform this by 
 
 Fi(xy) * f( x)F( xy)<i>( y) 
 
 and set dxi-=dx^r(x)^>(x) x=K(xi.) Xj,= ii(x) y^^(yi) 
 
 Transform now $ ( xy) in the same way i?ivin;? 
 
 $x( xy)=‘f(x)'^( xy)^( y) 
 
 Then we have ^ i{xy)= f ^ x) dlir f ^ J r ( x) F ( xt)^>it) f it) Hty) 
 
 ^^( y)dti 
 
 But from the value of d^i we have at once that this 
 
 = r ^^f(x)F{xt)^ity)w(y^dt=f(x)Fi^>(y) 
 
 X 1 
 
 That is ^^J,(xy)^r(x)^{ xy)^>( y) 
 
 It is evident no» that if is the result of a oerautable oo^nosition 
 
 By a orooer choice of functions we can reduce any function 
 
 P from the Jeneral font to a fore such that P(xx)^\, F xy) = dP^ ox, 
 
 P.( xu)-dF/Su and Fjxx) =■ 0 F,{xx) '0.; „ . 
 
 We now derive the condition of pennutability of a function 
 
 .tth a iiven function as the solution of an intejro-iif ferential 
 equation.; Let ^ 
 
 dif f erentiaf e as to x and as to y 4 ivin^ 
 iof which integral epuaUcn.- i as unknown we wiU let T. he the resolvent. 
 
 
 ■y 1 ' X 
 
 of which we will let f, be the resolvent • 
 .Therefore we will have froo these two 
 
 

 < 1 . 
 
 X 
 
 
 
 A 
 
 
 ♦ 
 
 4 
 
.Jf, 
 
 Hl3n 
 
 by subtraction we ellfflinate the unknown 
 
 I 'quatiop'.for f' 
 
 ‘ 
 
 and have an 
 
 
 ae 
 
 nte^ratin? the ri^ht hand side by parts we free the function ^ from 
 the differentiation as to and therefore have a froa 
 
 The solution given here and the final treatment are found in Foncticns 
 de lignes pp 61 et seq«,| 
 
 Let 
 
 FIFTH lecture; 
 
 Integror-dif ferential equations.] 
 
 We found the general form of integral equation by passing to 
 the llsi't from a set of n equations to be 
 
 f 1 [y(^) x] \^z(x) 
 
 1*1 
 
 Similarly we arrive at an integrordlf ferential equationby 
 the consideration of a sot of simultaneous dif fer??r44al equations. 
 
 jy„ z) 
 
 dy^/dz^F giui j/t '*t*f*! Un z) 
 
 I* r I* f r I* r p :• f i* r r p '* r r r r r i* i*r r i 
 
 dyn^dz"Fniyi Vn Vn z) 
 
 If welet the number n increase now so as to become continuously 
 infinite we find that we must substitute for the index a parameter 
 
 X giving us x 
 
 dy( z, x)^dz “ F\[y(z £ 
 
 which is an integrordif ferential equation.] We may also arrive at 
 equations of this character from partial differential equations^! but 
 the classification is not at all easily settled.] These equations 
 are solved by the methods of Cauchy (and earlier Laplace) called 
 by Picard unethod of suocBssiue aoof'oxifnat ion [See Coursatiji Hedrick s 
 Trans.] Vol 2 2 p.j 61 et seq.jl iGoursatiji Vol III Chap.] XXX) .] 
 
 The types of integrordif ferential equations studies are classified 
 under three typcs^i according to the o^araoterist io curves [See 
 Bvansj,! Co'Jloquium lectures p 90 et seq) i The types are the elliptic^ 
 hyperbolic/ and parabolic^ The entire fidd is an application of the 
 theory of f i o n f functionals (Evans) font tionel 1 es 
 
 Frechet),] functions in a Volterra (that is continuously infinite 
 
 

a uiil' er ’ o f 
 s 11 j I? e s t s ^ 
 
 iircensions) spac^, or t-he reporter of these led 
 in brief,, Volterra functions. 
 
 Derivative ani iifferential 
 of Volterra functions. 
 
 « 
 
 In orier to pusch our frontiers as far bacK as possible^ we 
 (Eust consiier next the ilea of variable Volterra function^ ani con- 
 sepuentl.y continuity of such function ani hence iifferential of such 
 function. A Volterra function ieoenclin:? upon f{x) is continuous 
 (Volterra sense) if when we substitute for fix) the function fix) + 
 where ^ix) is bouniei in absolute value,, then the variation 
 of F\{fix)}\, that is ^ j [ ^ (jc j f jcj 1 1 -S’|[rf.x:)]| -<o is infinitesitial 
 
 with t.- o is to be any arbitrarily stcall quantity. [There are 
 other iefinitions of continuity for a Volterra function^ for which 
 see Fonctions ie Liiines o 21.1 
 
 To iefine ierivative at for- F we p’”oc..,ei as follows: 
 
 choosing two values of x, one less than,, t 
 other greater than i, we vary the function 
 fix) between these two values, ^eeoin^? all 
 other values fixeJ,, the variations of the 
 function values bein^ less in absolute 
 value than t. The variations of F ^ 
 are then liviiel by o the ar«a enclose! 
 between the new curve ani the original curve, ani the littit of the 
 q.uotient founi. If such limit exists it is the ierivative of the 
 VolterrafunctioriFatE, 
 
 Liffi = f ' i]j 
 
 It can be usei to exo'^ess the variation of F thus 
 
 tF-^r^ F' [fix),l.] I tfix)'il 
 
 wnich is entirely anaios^ous to the formula for total iifferential of 
 a function of several variables, = ‘ * ‘‘’Fr. ' . ^e 
 
 ^lEay also iefine hij'her ierivatives, ani fini the re.markable tbecrem 
 |!that the seconi ierivative of a Volterra function at gri then is tn 
 ■■secQii ierivative at n ani then I, 
 
 fit? may now have as a Generalization of ouc orece 
 
- 15 ' 
 
 
 
 IQU ^ t i ciB i'^volvinJ "ierivgtives oT Volterra f‘uiictioil^3dt..',5;^' 
 
 
 J * * 
 
 1 id 1^ examole^, analogous to the equation 
 
 • T 
 
 _,, + • • ♦£naf’''3(/r. = 0 t'll *1 
 
 ph^se solution is U 2^ ^-•hn } ' ' ' Un^'^Ur,)- Constant^ that is ^ 
 
 ii General any function of zero orler ani homo neo li 'ao may have 
 
 as^he analogous fortii 
 
 f , x]\ ii( x) ix = 0 
 
 ^The solution of this is the Vol terra function 
 
 p 1 
 
 ‘ /’o'-.vCTOiM 
 
 'Ve may solve other forms similarly. A secon.i examole is 
 
 y iB ff’/d 1 + • • • .z/r. o d yn = 
 
 vith solution + - 'UP Miere =’ is of ie^ree ^ero. The integral 
 
 equation »ouli be f o h;" (i) ss shovel A thinq example is ieneralizei 
 
 from • 
 
 'dP/'dz^ll^F^^y iy( y t} ' ' ' } yrJ Uy ~ '- 
 
 which can be inte^rateh by first inteiratin-^ the sv^tem 
 
 i y 'i* 
 
 dyn ^ dz-ZuQn 
 
 fhe Generalized fOhm is 
 
 -oP'^dz ^ r,^p\[y{l),z^\ fo^a(xy)ym)d^dx - 0 
 
 A soecial value ter P' ^ives a soecial case 
 
 dy(xz)dz) To '•c( .xfi)f/(h)'^n. 
 
 1. -hp solution of the system above, which 
 
 To solve^this we ^ . e J , Solving these for 
 
 4. 4 ^ritinu iovui any arbitrary function o 
 
 the n constants ani xrtttnj xon, y here we 
 
 results we have the most uene’^al solutio 
 
 have first , \ 
 
 y(.x) ^C( x) o^( x^.\z) 
 
 e +->rtn Solvinu the integral equation 
 where G is an arhitra-,v function Golvi . 
 
 : for G we have ( r \ .r 
 
 Z(x) ^ y(x)-f z)y('^.)dc ^ ^ 
 
 ^ n fm-tion Any Volterra function of xJ 
 where ^ is the resolven function ■ .y 
 
 is the result iesi’^ed. nolications of these theories to 
 
 l-he-e are many aOO ^ Volterra ani his students. 
 
 Mechanics. ?or *e se consu 1 t t he memoi r s 
 
 '‘t.’Kv i 
 
 Liu jk * 
 

 ■ i 
 
 - 16 - 
 
 Latterly they have been applied to the study of Saturn ' sprir^fl*^ 
 
 nd to nebulae. 
 
 HISTORICAL.' 
 
 , y The first work done alon^ these lines be^an with Volte^ra's 
 
 thesis: Sopra alcuni problemi della teoria del ootenzial e Pisa 1833 
 
 In this infinite processes are considered.' In 1884 aooeared an 
 * z 
 
 integral equation of the first kind\, ^i>(x)=fo F ( ix,x)f ( z, a) da, in 
 
 •Sopra un probletua i el etro st at ica^ ' Rend.; Ac.; Lincei (3) 6 dp 315—318;^ 
 
 and in II nuove Cimeuto^ 16, (1834) 9 pp.; In 1387 the function of 
 
 line (Volterra function) aooeared in Teoria delle equazione 
 
 dif ferenzial e lineari, Soc.; Ital.‘ d Sci.; (Xl) 6(1887) 1C7 pp and 
 
 12 (1899) 69 DO.; There are other papers in the rend.' Lincei.' 
 
 In 1896 the method of studying problems of this type by starting 
 
 with a set of al.?ebraic equations, whoxe number is then made 
 
 continuously infinite, was first used in Sulla inv-.'^^ione de^li 
 
 inteifrali defirti, Atti Ac.. Torino 31 (1896) 311-523^. 400-408., 
 
 557-567., 693-708. See also Rend.- Lincei (5) l77-18o^ 5* 289-300. 
 
 A summary of preceJin:? results and new results are to be found in 
 
 Annali di matematica, (2) 25 (1897) 139-178' Inte^ro differential 
 
 equations were studies in Rend Lincei 1909.; Since then many oaoe*s 
 
 have appeared Lfp to 1914^ when work was suspended for the WAR. 
 


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