} re E vives wee : ‘ Mv . ray - rer oS 8 q NS AARC oak SG ; . . eet ae ‘ i ee Ny ae ; ih RAAB AA See te : pon Aa N revtoteetny 0 gh ee CRIES So, . ees ye ee SER RAE aes . : AR ARALY SY RUN LS Pa Sh ce Qeyn SUA a hee ES CaS Ay Cee a a hy oe Donon ‘ ose : . . 5 7 of we 3 ; ' . : bara Shey H ; F SSSNAR ees t : naa se SESS Ge ae : . < SOLON NEN eta orare ee tety E : BAN Sh eri SEEN aga ‘ ¢ . Wreteeatety hee “ey : ‘ BERET RGR Cea) t ; cite MS SARA . ; , : AS aa Ae hs i*-< e . . ey awe She tne ae : aa eatetete ely tesa MeL Bees LD OEP ANDO Stan meee 7 aS atyraticite wat bat ’ . H 7 , CeO Sete So i ‘ : eet a ese L Leena { . LAREN SS ites » eyes RANEY 6 estes eRe : BRUNET) Sth TAS ee MUR E a vse RRA tek Wiad ts Rod MAS : : oe . A Bie as petri Sh See on Rees ie eee it . 7 tees : pie tara raat Nec Te . : 7 area ARNIS Non Gace eet ae eee oa : . Noho s8 WES ARSE REET Sa ca : were \ Cibo Ne . . . Dt. eS ats ae : F 3 . . ae beeen mts : : . . . res hae F wanker . . . 2 re a * . we he} wim te o PVA if eer Lees a arene - ris r Roa t Plud A Rete er eee ysis anette, ‘ > \ s wy NRL te teat ° . ee . ., Walales. gustatatigel ae aes) rs 4 wate 3 aN fat beets t ase 3 Pre RANE _ ne ae : . sa +Y * SNES ST . . - . 7 . 7 arr A? : eum _ PEPE Pe PO PES OB Pe } 3 Pee i : é : . ssgh aa of! tee mt ‘ ’ : a? 8 sacaconcsnasscgceusnicect eters cee conecevenctotre seams setae cae 36 On Relations among Perpetuants. By A. Youne, M.A., Fellow of Clare College, @amibrideetensccsece ccoeene soe Sapispesecas|sienewcs Gucane aens deacetaceniseeesteeseenscce scent meee cace 66 On certain Quintic Surfaces which admit of Integrals of the First Kind of Total Differentials. Second Paper. By Artuur Berry, M.A., King’s College, Cambridge 74 The Law of Error. Part II. By Professor F. Y. Epcewortn, Oxford. Communicated bya Garba, MathewssmiinAte WES 8 tec: saememebstt resem asaman chert neces cen es-peccisan eens 113 Memoir on the Orthogonal and other Special Systems of Invariants. By Major P. A. MacManon, Sc.D., F.R.S., Hon. Mem. Camb. Phil. Soc................... Brae ioc 142 A comparison of the results from the Falmouth Declination and Horizontal Force Magnetographs on quiet days in years of Sun-spot maximum and minimum. (From the National Physical Laboratory.) By C. Curer, Sce.D., F.R.S ............ 165 The influence of very strong electromagnetic fields on the spark spectra of (1) vanadium and (2) platinum and iridium. By J. E. Purvis, M.A., St John’s College, Cam- [DUA FRE qentascecr ao bode pene 105 060A Gano CRAEiene JRO HOD EE MOFE con Cesc GasaRER REA Ondccn< cop soneapacan enBue 193 On the Asymptotic Expansion of the Integral Functions 2 a" T(1+an) 2 a" T(1+n6) nao U(1+n) ond aa (l+n+n6)° By E. W. Barngs, M.A., Fellow of Trinity College, Cambridge ..................... 215 A class of Integral Equations. By H. Bateman, B.A., Fellow of Trinity College, (Chim DUD ccoadedoddusecte s nossnondaesBe BBG ye oTEdbon sno adsEmEmpds6 Sse cesabencnonedesnanskabe sttahine 233 vl On Functions defined by simple types of Hypergeometric Series. By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge .............:scee-sseeeeeeseeesenss eee eeeernees The application of integral equations to the determination of expansions in series of oscillating functions. By H. Bateman, B.A., Fellow of Trinity College, Cambridge The Variation of the Absorption Bands of a Crystal in a Magnetic Field. By W. M. PAGE Bb PAty Kan o-sy Colleges Cambri Semis err reetscte lets etetee late ete eee eee rears On the Asymptotic Approximation to Functions defined by Highly-convergent Product- forms. By J. E. Lirruewoop, B.A., Trinity College, Cambridge ..................... The Reality of the Roots of certain Transcendental Equations occurring in the theory of Integral Equations. By H. Bateman, M.A., Fellow of Trinity College, Cambridge On the Solutions of Ordinary Linear Differential Equations having Doubly-Periodie Coefficients. By J. Murcur, B.A., Trinity College, Cambridge ....................0005 PAGE Absorption bands of a crystal in a magnetic field, variation of 291 Asymptotic approximations 323 Asymptotic expansions 215 Barnes, On the asymptotic expansion of the integral functions 2 2"T(1+an) aT (1+n8@) a15 >> = noo I'(1+2) nol (1+n+n80) Barnes, On functions defined by simple types series 253 Bateman, A class of integral equa- tions 233 Bateman, The application of integral equations to the determination of expansions in series of oscillating functions 281 Bateman, The reality of the roots of certain transcendental equations occurring in the theory of integral equations 371 Becquerel 291 Berry, On certain quintic surfaces which admit of integrals of the first kind of total differentials 74 Bessel functions 270 of hypergeometric Chree, A comparison of the results from the Falmouth declination and horizontal force magnetographs on quiet days in years of sun-spot maximum and minimum 165 Crofton, Morgan, law of error 45 Double zeta functions and double gamma functions, expression in terms of elliptic functions 1 Doubly-periodic coefficients, equations with 328 Edgeworth, The law of error 36, 113 Electromagnetic fields on the spark spectra, the influence of very strong 193 Expansions, asymptotic 215 Equation, a partial integral 246 Equations, generalised Picard 389 Equations, Halphen 426 Equations, integral 233, 281, 371 Falmouth declination and horizontal force magnetographs, results from 165 Fourier coefficients, equality 177 diurnal in- GENERAL INDEX Fredholm, 234, 281 Functions defined by simple types of hypergeometric series, Part 1 the series ,F, {a; p; x} 254 Functions defined by simple types of hypergeometric series, Part m the function oF, {p; x} 270 Functions, Bessel 270 Gamma function, simple 7 Gamma function, double 20 Halphen 426 Hardy, On the expression of the double zeta function and double gamma function in terms of elliptic functions 1 Highly convergent product forms 253 Hilbert 236, 281, 373 Hypergeometric series 253 Integral equations, a class of 233 Integral functions, asymptotic ex- pansions of certain 215 Invariants and covariants, irreducible 151 Tridium 213 Kummer 270 Law of error 36, 113 Laplace, law of error, 51 Littlewood, On the asymptotic ap- proximation to functions defined by highly-convergent product-forms 321 MacMahon, Memoir on the orthogo- nal and other special systems of invariants 142 Magnetic field 296 Magnetographs 165 Mercer, On the solutions of ordinary linear differential equations having doubly-periodic coefficients 383 Ordinary linear differential equations haying doubly periodic coefficients, solutions of 383 Orr 256 Orthogonal and other special systems of invariants, memoir on 142 Oscillating functions 281 Page, The variation of the absorption bands of a crystal in a magnetic field 291 Picard 372, 383 Platinum, 211 Pochhammer 270 Perpetuants, relations among 65 Purvis, The influence of very strong electromagnetic fields on the spark spectra of vanadium and platinum and iridium 193 Quintic surfaces which admit of inte- grals of the first kind of total differ- entials 74 Quintics with a double conic and a triple point 75 Quinties with a double conic and a double point 76 Quinties with a non-degenerate double conic but with no distinct multiple point 83 Quinties with a double conic, consist- ing of two coincident straight lines, but with no distinct multiple point 105 Quintics with a double conic consist- ing of two distinct intersecting straight lines, but with no distinct multiple point 90 Series, hypergeometric 253 Spectra, spark 193 Sun-spot-frequency 166 Syzygies 157 Transcendental equations, reality of roots of certain 371 Variation of absorption bands 291 Voigt 322 Weierstrass 380 Wolf 186 Wolfer 166 Young, On relations among perpetu- ants 65 Zeta and gamma functions, simple 3 Zeta and gamma functions, double, 13 : Zeta functions, double, special cases of 21 Zeta and gamma functions, connec- tion with Barnes’ contour integrals 31 CORRIGENDA AND ADDENDUM. . 36, line 13, for “hk”, read “ky”. . 42,1. 9, for “as unity”, read “as of the order unity ”. . 43, 1. 8, for “function for y as the first”, read “function of « for y, the first”. . 51, last line, for “RAx”, read “R”. . 52,1. 7, for “S”, read “ So (E)”. . 53, 1. 6 from bottom, for “ | Drea ot ea 0 . 60, 1. 9, put brackets outside the right member of the equation and outside the bracket on the right, “yo”. . 60, 1.10. Make similar correction. G1S 18; OMe k> 1 eae Coxe teen. . 118, put at the beginning of the last line, also of the line fourth from the bottom, and the line seventh from the bottom, “ — ” (the minus symbol). . 119. Make similar correction on lines 3 and 5. . 123, 1. 10, for “in general”, read “ at first”. . 123, 1. 11, for “will be found necessary”, read “is convenient ”. p. 141 Add ‘The writer desires to refer to his paper on “ The Generalised Law of Error” in the Journal of the Statistical Society for September 1906; where a condition which is mentioned only incidentally in the paper on the Law of Error in the Camb. Phil. Trans. (at pp. 114, 115), viz., the case in which the series of coefficients k,, k2, #3... descends less rapidly than by powers of 1/ ——, BG ) 2. (a+ mo)’ where (a+mw)-* has its principal value. We have now to pick out the term independent of n in the asymptotic expansions of (8) and (4). § 3. For this end we define a series of functions ; d k+1 Ae (U— a) = (=) A, (u— a), d Ay (u-—a)= A (wu — a), Kee) elt a(u=a)=—} - y(ua)dus | ae (5), == 2 foe TOO) 5 ih w | @ . roo el As (u— a) = -| A, (wu — a) du, u the path of integration being in each case the straight line (ii). We also define a similar set of functions M1 (W— A), v0) fo (U— A), fy (U— 4), «. in the same way, using the straight line (i). Evidently all the p’s with suffixes $1 are identical with the corresponding 2's; but lent es z it cidsdiueyoe (6), 7 u-—aj)=—— ba ( ) on \ and so on. It is easy to see that every X or mw vanishes exponentially for w= ae or w= 00 eX, Suppose now c=—S, and 0=r and the right side to 1 : Thus we find log 1'($)=4 log z, T()=7, which is known to be correct. § 7. All the formulae obtained so far are valid only for a restricted range of values of a. I shall not now discuss the modifications which have to be made in them when a lies outside this domain, as it is not this set of formulae but another rather different and less symmetrical set which I shall require later. A Second Set of Formulae for the Simple Zeta-function. § 8. It will be convenient to adopt at this point a notation more natural when we are dealing with functions connected with two periods. I shall denote by w, the quantity up to now called », and by @, another quantity satisfying the conditions R(w,)>0, R ee >0. I suppose that a les within the strip bounded by lines from 0 and @, parallel to the positive direction of w,, and I consider the complex integral the contour of integration being now the rectangle 0, (rn+1l)o, (w+1la,+he,, ke, (k being a positive integer) modified as before by a small are round 0. The value of the integral when & is made very great and the are very small is N ws i (una Ss — | \ 2 La(u—a)du 0 0 wo FF, (u xe Ny DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 8, N being equal to (n+1)@, as before. As in § 3 we can prove that the term inde- pendent of nm in the asymptotic expansion of the second integral is 20 tp du i) Ay (u— a) 3 it remains to discuss the first integral. § 9. For this end we define a new series of functions d\ka gi, -~(U— a) = (5) A, (u— a) d,.(u—a)= (%) A (wu — a) Bong npEenaceccar (20), gi. (U—@) =A, (w— a) + CL, u d,2.(u—a)= / A, (u—a)dut+C,,(u-a)t+ C2 characterised by the following properties : (i) each is the derivative of its successor ; (ii) each is periodic in @,; (i) each satisfies the equation ["s (u) du =0, Jo in which the path of integration is supposed rectilinear. This is certainly possible. For suppose that functions up to ¢,,,(u—a) have been defined in such a way as to satisfy these conditions. Then ivi (U+@,— 4) — by (U—-a) = | ‘S oi,» (wu) du ~u—-a SO du 0 (as ,,,(u) is periodic) =(). and so $,,1:(u) will be periodic, and by choosing C,,,, suitably we can ensure that it also satisfies condition (iii). It is easy to calculate the first few functions ¢,,. Thus C,, is determined by the equation a (z) cosec? Beso) +0; | du =0, @) ; } 0 @) so that C,,=0, and bya (u—a)=—* cot ™= 4 Otis, Ge adoRbonosatasonnocoubacdE (21). 1 1 Vout. XX. Part I. 2 10 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND cot JQ @, @) = L | og sin mus |" peers pS ore T(u—a Similarly = ee | SU A v) ahh 1 @; 0 where k& is an integer. We can determine & as follows. Let vine ro =a+ia’, a’ >0. @) . T(Uu—-a aa : Then log sin a ) = log sin (w — a—72’), 1 where z is real. The imaginary part of this is a {( tanh a’) 24mm — tan) —— see l tan (x — a)) where m 1s an integer, and Se tanh a’ za ——~< tan 2 tan(w—a) 2 tanha’ . : : As x goes from (0 to 7, tan ——— increases or decreases by some multiple of 7. tan (7 — a) This must be a negative multiple as d _. tanh a’ tanh a’ — tan? — 5 =—-= = <9, da: tan (2 — a) sip? (z — a) + tanh? a’ cos? (# — @) and it must be —7 since tan(#—a) can only vanish once. Hence Og chases saanpe ec Re (22), @) and ob), (u-—a)y=— log sin 7m (u SE — a)+C,,, wc c cece cere eeeeease (23). : @ The branch of the logarithm to be chosen is at present at our disposal. Again C.3= oe | % diee sin GSD) ae (u— ao} du : ( l 7 @) @, MJ 0 cide le yp . w(u—a) = — (4@, — a) — + — log sin Ve du. ® @J0 @) To determine the value of this integral integrate [ . TU | log sin — du, : @) round the parallelogram (0, —a@, —a@+a,, @,). The points 0, @, may be avoided by small ares of circles, and we choose that branch of the logarithm which is real when wu is on the line (0, @,). This side contributes rw 1 eaTz2e a, [™ . —| logsin—du =-—— | log sin «da = @, log 2. J0 @) 7 Jo = DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 11 The first and third sides contribute azz, and the last contributes cn . 7(u—a | log sin Nee) lu, Jv @) which is therefore — w, log 2— ami. 7 ant : : Hence C,,; =— (4a, — a) — — log 2 —- — =—log2— dart... eee (24), @, @, 7 (w—a) provided the proper branch of log sin be chosen. This is the branch which is @, real along the straight line (a, a@+q@,). Thus finally . w(u—a) 7m ; ts $r,3 (u — @) = og sin 7 "= +—(u—a)—log 2—477...... ee. (25). @, @, - § 10. Suppose now, as before, =— $, ; 0S7r ——. S( y m=—-k (a a5 ma, )* Hence (27) becomes —1 il an m=-k(@ + mw)§ — 2m 16, (8, a) + pas64).649 | [by ron(u—airr du 720009 ...(28). =| Arts (U— a) UST aw | 0 It must be carefully observed that the hypothesis that a R ca >0 1@,/ . ee ; oe a : : : is still essential; if IR (<)0, K(w,)>0, 2 (2 ) =0 y - : ion i 2m, 2p. Various complications, interesting only from the point I shall adhere to these hypotheses throughout. 14 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND taken round the parallelogram 0, Pi [=(pu +1) e), Pi +P. [= (pat) o], Pr. The origin must be avoided by a small circular are whose contribution to the integral can be made as small as we please, and w~* is to have its principal value. It is easy to see that the residue of @(u—a@)u** at a+ me, + io, 1s Ue eS) (a+ m,@, + M.,)°* and so, by Cauchy’s theorem, P, 1 1 rP. ia eens [e@-o = > aS du— | p@—O\5- GER] See) a ? | m,=0 m,=0 (a + M,@, + Ms@»)' § 15. We now define a series of functions gx. (u—a)=—* (u— a) | d (u—a) = —' (u— a) $:(u—a) =e (u—a) + ee Oo (32), go. (u—a)=— C(u—a) + Ci (u—a)+C, ds (wu — a) =— log o(u—a) + $C, (w— ay + C,(w—a) + C; to satisfy the same conditions as were satisfied by the functions ¢,,(w—a). ‘The analysis of § 9 shows that this is certainly possible. Then if 0< R(t) <1 py: du ah dy (U = are is convergent, by the general theorem already quoted. § 16. Now let us consider the integral i ( du I(U— a sie: J0 s wv Suppose that —s=o, and r For the argument of the preceding paragraph shows that the finite term in this expansion is the same as that in the expansion of P, (-)"(o+1)co...(a—7r) | bres (U— a) (w+ P,)7 7" du, ~0 and it may be shown as in § 3 that this tends to zero for n= a. § 17. We-therefore find by equating the finite terms in the expansions of the two sides of (31), — 2mif.(s, a, @, @2) =s(s+1)...(s+7r) | rata clas sche (33), * Dw. | rear —ayus dy — | Wris (u — @) da ) J 0 where ¥,,, is one of a set of functions y, formed in precisely the same way as the functions ¢,, except that in their formation », plays the part of @,. § 18. This formula expresses the double Zeta-function when & (s) < 0 and is not integral. To obtain formulae applicable to other cases we have only to differentiate with respect to a. Since = (Ss, d, @,, @:) =— sb. (s+ 1, a, @,, w2), rs) : aa gd, (u— a) = — dy (u—@), we find by differentiating « times D(s+«) el ey COE Dd reed ae aah Fe ndasaseee (34). = Pras (U— @) UI du =| Wrign(U— a) mesg 0 0 § 19. The most interesting case, as being that from which we shall derive the formula for the double Gamma-function, is that in which 0 < R (s)< 1. If in (34) we suppose —1< 3 (s)<0, r=0, e=1, r+3—«=2, we find "Dw, Fwy —27if.(st1, a, a, @) =| ds (u — a) uw *Tdu — / ro (wu — a) udu... (35), 0 /0 or, writing s for s+1, so that now 0 < %(s)<1, — 2771f,(s, a, @, @2) = [bau ay _ hee be (u— a) = Seale cere. sinads (36). 16 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND Since b.(u- a=" {b; (wu —a) — ds (—a)}, we obtain on integrating by parts — 2trif,(s, a, @:, @o) (0 <4 (s) <1) - z wey OE OO agg ae (37). =s [ {b; (u —a) — $; (— a)} =e -s [yrs (u — 0) =e (—a)] 2 § 20. This last transformation may be generalised, and further formulae obtained, by the help of Cauchy’s notion of ‘intégrales eatraordinaires’. Suppose that f(x) is a real or complex function of the real variable « which is continuous with all its derivates for all finite positive values of #, and can be expanded in a Taylor's series Ay t+ e+ a a+... valid near 2=0. Then if m+1l an [ur @)- a, — ... —(m—1) ana" } eee Re 28 0 ( oom t—2. NE FSO Ih at If m=1, 2 log (1 — ¢")=—- == oon ey so that C,=— $7 + flog g— Ay log A cscs cs ccea: sesceeceeesensseneeee (46), if suitable branches of the logarithms are chosen. Thus oi (u—a)=—(u—a)+ Wer ais sch iodectd ache ooo (47), uh ee ee aay a Rt (48), $a (u — a) = — log o (ua) + 5 ra pe pb 4) — dri + flog — ylog A...... (49). * Halphen, loc. cit., t. 1, p. 428. + rade. Cours @Analyse, t. 1, pp. 426—7. DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. nS § 23. It is easy to see that the constant C// in y, is @, SOOO IROL SOOO CORSUCOHTONUCTC OOOO OUOUS EOD OO0C (50), and that of =-= =oebece bee ehae aS. eo (51). The exact specification of Cy is more difficult. It follows from § 22 that d; (u— a) = — log o¢ (u—a) + $0, (w—aP? + C.(u—a) +, n - T(w—Ga) Hy ae He Pea ss 2 Gg" _2n7 (uw — a) =— log sin ears & + Der, (u-—ayP+ = (uw —a)—log 2 57 Tp ae) cos eT ae 7 (u—a) where that branch of log sin - is chosen which is real on (a4, a+@,). Also @) vw; (u — a) = —loga(u—a) + $07 (u-—ay + CY (u—a) + CY, where the value of C, depends on which branch of logo(uw—a) is chosen. We may, for instance, agree that that branch is chosen which is obtained by going from near 0 with the branch chosen in ¢, and returning near 0 by a loop round «@ nearly coinciding with the line (0, a). The value of C,’ is then fixed. To find its explicit expression would, I think, be troublesome, as we should need to use ex- pansions for elliptic functions in terms of rie, =e %, which are not given in the books, and cannot be written down at once from those in terms of TI q=e*, @ - because RW (=) <0. Fia. 1. However, it is not in any way necessary for us to know an explicit expression for Cy’. It is evident that if we make w=0 in @,; and wy, the corresponding values of log «(—a) differ by 277. The Double Gamma-function. § 24. I shall now deduce an expression for the logarithm of the double Gamma- function by means of the equation* rT, (a, Q@i, @2) ) | 52 log Gas 8 [as 2 (S, @, @,, @2) eg 1s be (52). By (87) (G19) = — 7it, (S, a, @,,@2) (0< R(s)<1) Bey d ens du =s| {b3(u— a) — };(— a) — s|. labs (u — a) — Wy (—@)} sae * Barnes, loc. cit., pp. 330—334. 20 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND 7d ah du Now a5 lef, {bs (w— a) — $3 (— a)} as | | ae du a) ice Sail .. (58). =| ips (wu =O) ps (ee a); ys al s| {hs (w - () = db; (- a)\ ys | : 0 Each of the last two integrals becomes divergent for s= 0. Let h be a small fixed positive quantity. Then ee {p;(u —a) — b; (—a@)} gs = ie +f) {hs (w— a) — b; (— a)} — ie du ag du _ bs (— a) {s(u=0)— bs(—@)} eat] bY) eS Chany Each of the integrals on the right is continuous for s=0*. The last term may be expanded in the form — $;(— a) — log (ha,) + = > [= b;(— = me Lye ae | ‘1+ _ Now ae aCe a =— dy ( a)log (~ — and a i 3 (— a) {| —— --1)% wae = ;(— a) log (1 + ha). 0 a +t Hence ie id; (u— a) — b;(— @)} du _ ue O4 |” ‘|9 Gaye os (— ee du ys “ +u)u plus terms which vanish for s= 0. =| tose —@)— 6-0) ER se u 1 S _ $3(— 4) toa ore Ano 3 plus terms which remain finite for s=0. Hence by (37) and (53) Wy MOn fakes S d somal SiN ee - — 2ri log ae ay" 2) =|. | ae -a)- 80 I} <— th a —a)- a . at 2 gene (54). — u ay Again ke {bs (u— a) — d(— a)} + [“"s. ob; (u—a ee We can at once deduce formulae for the various logarithmic derivates of the Double Gamma-function. For if, in Mr Barnes’ coed Wo) (a, @, @2) = - log TG; @y, Oa) teaeeceeecee theres oe (55), we obtain at once : ee ds—m (= @) (—a)) du _\ymti (m) = = (—)" Qari,” (a, @1, @) 3 \6 s—m (WU — a) — re as Me eo - (56). aes _ gq) — Yen a) du iE em et 1l+u 2 * Here, as in § 12, I omit the proof. This remark also number of theorems concerning the uniform convergence of applies to the differentiations under the integral sign in certain classes of integrals which enable us to justify the this section and elsewhere. I have proved elsewhere a assertions of this kind made in this paper. DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 21 Special cases of the Double Zeta-function. § 25. We have now found formulae for the double Zeta-function for all values of s for which 3 (s) is not integral, and also when sg is a positive integer >2, in which case (Ss (s—1)! € (8, a, @, @2) = For s=1 and s=2 the Zeta-function becomes infinite, the finite terms in its expansions in powers of s—1 and s—2 being respectively — yr." (a) and w,” (a). There remain two cases to consider, that in which s is complex, and i (s) integral, and that in which s is zero or a negative integer. By (33) of § 17, if Of r would lead us too far from our present subject. We have now found formulae for the double Zeta-function for all values of s save s=1 and s=2, for which it becomes infinite, and y,”) and yw.” take its place. A Formula for log p:(@,, @s). § 26. We can obtain a formula for log p.(@,, @:) by means of the equations lim flog D(a, w;, w2) + log uv} = 0, a=0 and (54). We require to consider the behaviour of hens! —«) du -{" i. uU—@ aia es $s( )- = Se u 0 J he, when a@ tends to zero by a a lying entirely within the principal period parallelogram. Under these circumstances (see § 11) ¢;(u~— a) tends to a definite value for all points on the axis of @, except the points 0, @,, 2@,...._ These limiting values define a function ¢;(u) which becomes logarithmically infinite at @,,2,,..., but which is easily seen to possess the characteristic properties of ¢,(w—«q), in that it has , as a period, and "%. (w) du=0 It is not difficult to prove that 2% oe —®) ay hw, is uniformly convergent for a range of values of a including 0, and so continuous for a=0. Thus ew, paw h | dee ‘bs(u) 4, (— a) log ; =e ? if we neglect terms which vanish with a, as I shall do without remark in what follows. ~ he, hw * Barnes, loc. cit., p. 273. 24 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND ho, hw, Again | = $; (—a) log (1 + ha,)—| {log (uw — a) — log (— ay} 0 0 “he, , du + {hs (u— a) + log (u — a) — ds (— a) — log (— a)} re /0 log (u—a), log(—a) being chosen so that lim log (uw — a) = log (— a), u=0 and lim {$,(— a) + log (—a@)} = C,. a=0 The last integral tends for a=0 to hw, | (Ps (u) + log Ure C;} = : ~ 0 We may therefore replace the original integral by ie bs (u) + $3 (— a) log (he,) he, hw, hw, l -| fleei@= ay lop (ae: | {;(w) + log wu — one 0 U Jo u which is easily seen to be equal to 200, log u— C,) d hes d ik {es(w) + a ‘ “+ #:(—a) log (hie) — | ~ {log (w —a)— log (= a)} | Soe =| ee Es lt+u ps ho, U(1+u) We may replace $,(—a) by —log(—a)+C;, and then C, may be omitted except in the first integral, since du te du he, log (ha) — i 1+u uk itn u(1+ uu) : as appears on working out the integrations. The integral along the axis of , may be treated in the same way. Hence the right-hand side of (54) becomes O00, { log wu a. Cs) du \ i, ke aaa he ee Log u—C,) du -{~ {yr sy) Tu = UU = log (— a) log (hw,) + Log (—a) log (he) ey -| (log (u —a) — log (— a) 0 liws + i (Log (uw — a) — Log (— a)} — a ( hw, log ua i Lo 4 + ik = So du o 1+ te [ l+u -|"™ log w x fie Log u ae du a ey OGLE Was, (l+u) ) DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. bo ol In this formula log and Log denote different branches of the logarithm. The last four integrals are equal to hws / 1 log al . a ie (1 a5 -) ear du — 2m log (he), (where the integral is taken along the straight line from ho, to hw,, and m is an integer) = — {log (ha,)}? + $ {log (ha,)}? — 2ma7i log (has)... 0... cece eee neces (65). Now let us consider the integral fi 4= 8) du } og (= ae Uu taken round the contour C shown in the figure, in which 0,, 0, are points nearly hwo Pa A hi : (2) 7 N 4 aS (c) Ca x Op. uw NS —— 0; ae (1) Tao, Fic. 2 coinciding with 0. We start from 0, with log {(w—a)/—a} nearly zero. When we get to 0, its value is nearly 277. Hence the contribution of (1) is nearly equal to the integral in the fourth line of (64), and the contribution of (2) exceeds the fifth line of (64) by a quantity nearly equal to ; [ hes dy — 2arr = Jo, U j The contribution of the loop is practically ie du Jo, U : Hence, by Cauchy’s theorem, we may replace the fourth and fifth lines of (64) by hog u—a\ du . pre du log | ——— } — — 2mm -—- u Ie hw, MS u = {log (hw,)}?— $ {log (ha,)} — log (— a) {log (hw.) — log (ha,)} — 271 log (h,) + 277 log a, neglecting terms which vanish with a. The sum of this, (65), and the third line of (64) is 2riloga. There is apparently an additional term, viz. an undetermined integral multiple of 27rilog (hw.); but this must vanish, as the result cannot contain h. Von XX Pan I ii 26 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND Henee, finally, 2m log ps (@,, @.) = lim |- 2rt log = — 2mi log a + 277 log aD’, (a, @,, | a=0 2 1) 2. : er: oF | ...(66), Jie =e logu—Oy) du | =|, 1s (u) ies We i Ya Cu) + 1 pe \ a it being understood that the right branch of the left-hand side is selected, and that the logarithms under the sign of integration are so chosen that the limits of ; (u) + log uw — Cy and (wu) + log w— Cy for w=0 are each zero. § 27. I shall next find a similar formula for the modular constant y2(@,, ) defined by the equation Yoo (@,, @) = — [ae (@, @,, @,) + ‘| Boon nastododconcdhoddonnese (67). a We found that QriWrs” (a, @,, @) = Ow, de (= 4)) du — [er he (w— a) — Ya(= a du \ veay = [Pe #) Wee | Ty l+u su 0 Now it is easy to prove that ie |? (w= @)— BED} & 9, ( (— ) 1 pea Me [ee — Fh ee gun) 6.0) = ho) wT) ; ( a ee —a)/ ‘$s (= @) + {; (ho, — a) — b;(— a) — hard, (— a)}/ho, se) ea du, ow; wow, du : ; 28 2 (— a and iB =|" IPs Ce W a) a $s rs @)) we w Ps (ons 7 @) a Ps Ss a) Me = ho, ee du, and the sum of these two expressions at once reduces to the form required. Hence l+u Ua urs (— «)) du Ll+u w | Qarivs") (a, @;, @2) = ie |¢. (wu — a) — d;3(—a)— eo} | = ten hae =a) — Any) J 0 § 28. Now Pebacu =U) = ps (— a) — ups (— ay} = hw, = | {.(u—a)+ log (u— a) — C, — C,(u— a) — $3 (— a) — log (— a) + C, — Cra /0 S Dh ay CE A | ; ul du uds(— a) + A ~ et Tile I, les (w— a) — log (— a) + aloe rho : : , hwy = {d; (wu) + log uw — C; — Cru} a = | | log (w — a) — log (— a) + ‘| < : Jo uc ( ; a) w DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 27 if we neglect terms which vanish with a. Hence 3a ea u du Tw, pha, =| {; (u) + log u— C, — Cru} — a t+ $e (—a) log (1+ he,)=[ {log g(u—a)—log (— ale das 0 a} Ww : cow, ( > (— 7 Again ie eae a)— woe eat du ,(—a) 1+ ho, =|. eo hw, Se a)log (~F5-"): Be Se ere eee) Thus Flee -2)-6(-@)- “SS _ pee Cru) du =) {os (x) + log u—C,— Tae ‘2 _ $s (—4) + $2 (— a) log (ha) ha, a ( s Cru ) du = ig iles UST Tau) we a hon Ci du o l+u du an fas {log (uw —a) — log (=a) + & a We may replace ¢.(—a) by ac, and ¢$,(—a) by —log(—a)+C;. Then it is easy to see that C, and C, may be omitted except in the first line. Hence Qrrivrs”) (a, @,, @s) a | Cyu \ du ue — A Cw) de Ee I je tlogu-C— pbe Ss yr Ov ) du -[ cw) + Log u— Cy — an = \ log(—a) _ Log(—a) | | | | 1 } cs a {log Geen) — log (hes)} + ho, R@g > eekenreee (70). fH ag [a ho, we hw, ue wy du \ we Tre, - {log (w—a)—log (a) +7) 0 +[" {Log — a) — Log (—a)+ ae aah We now take the integral [ flog ee —a)— log (- a)+ 4 28 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND round the contour C considered before, replaced by We find that the last two integrals in (70) may be hw. } , hwy _ f% i Neate —a)—log(—«a) + “| ah _ 2m { ae + ani [ a hw, | m Jo, UW 0, W hw, , a er, = || a ae log (— a) in log ( a) 1 | hw, C he, a) U / ho: 5 (fe 1 ho. a a8 i wat ( % ) ; Tic, (0 w, Dim\ |¢ (Cc) aaee \ wo hw, u hw, And since (by Cauchy’s theorem) we obtain finally by reasoning similar to that at the end of § 26 : , 1 — 2071 Y22(@,, @2) = 270 Fa (a, @,, @) + | a=0 he |. (u) + log u— C; ae 0 Ek el Peep OCUCOU OCC (fal)y. Llu) wv um * Berl ead pS. Cyu) du IL [Psy + log u— CO; iseu are § 29. If we differentiate (69) we obtain =, 2rriyp,”) (a, @,, @) Se Fe at er ean UI) ae =f Lo.(u-a)= g(a) — OO} -[ % fy (u SN = (Si uw, (— a)) du l+u Jw By a transformation similar to that of § 27, = Qari,” (a, @;, Wo) = 2{ \¢ (w— a) — $;(— a) — ud, (— a) — zu? gu (—a)) du 1 pu fe) eee (73). Bai ea) 4.0) wifey ee ; hae a) — Wr; (— a) — ups (— a) ca 7B And by analysis similar to that of §§ 26, 28 we can deduce Qariry¥, (@, @) = — Qari ka (4, @,, @) — sl. a 40) du =2 4 (2 o - 0,— a z Se aerogoonobdouecdoanc ; I \ (w) + log wu — OC; — Cyu Te “| aay (74) 20, 1 = 2 jon +log u— CO, — Oyu — Gu 0 1l+u) wv n bs ) . a . . . . Thus Mr Barnes’ three modular functions- are expressed as definite integrals containing elliptic functions under the sign of integration. DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 29 General values of a. § 30. We have so far restricted ourselves to the case in which a lies within the principal period parallelogram, Now suppose that a lies in the parallelogram kyo, + ks, (ky + 1) a, +h,@., (hy + 1) @; + (hk, +1), kyo, + (ky + 1) @s, (k,, k,>0). Under these circumstances the right side of (31) must be replaced by pyn—k, pan—ky 1 —2ri(s—1) > CS ———— ———— ‘ ) -k, ky (€@+m,@, + m.@,)* pyn-k, pon-k, pn-k, pyn—k, —1 pyn-k, pyn-k, —1 -1 -1 Now Si eS oS. SS —k, 9 0 -k -k -k -k, -k 0 0 The constant terms in the expansions of the first three sums are respectively =} ‘| (s, a, @), w.), > & (s, a@+m,@, @»), —ky ck and = f(s, @+m@,, ). —ky =i Thus — 2m E (8, @, @, @)+ DY O,(s, a+ 7m, w:) =k Se Sars 1 as = g (s, @+ M.@,, @,) agen! =) (a +m,0, mar ee: (75) = 5(st 16+] [dees (u— a) du _J0 — eay Ales (uw — a) ee du] 0 (0Sr Far re ee he m,=—kz ; 30 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND Summing these expressions from m,=—k, to —1, and from m,=—4, to —1, and sub- tracting from (75), we obtain an expression for =i oi Par xc aes) See See | are tee (77). =k, -k, (A+ I,@, + Me.) We may also, if we like, express the algebraic terms of (77) as integrals along the axes of w, and »,, and so obtain a formula for — 277 €.(s, 4, @,, @) simply: but the formula is complicated. Of course formulae for lo oe for general values of a, may be deduced. There is also the case in which a lies on the other side of one or both axes to consider; but my present object is only to indicate the way in which the formulae of this paper must be modified for values of a@ other than those especially considered. Special Cases. § 31. Special cases occur when a is on a side or at a corner of a period parallelogram. In such cases we may proceed as in § 12, if the special value of a is not one for which the function which we want an expression for becomes infinite. The most interesting case is however that in which a@=0. Suppose 0<4R(s)<1. Then by (37) ow, {bs (u —«) = db; (— a)} a => | {yrs (u —a)—v; (= a)} a © — 2 €,(8, a, @, @:) = 8 | “0 Suppose now that a tends to zero by a path lying inside the period parallelogram. Then it may be shown by an argument similar to that of §§ 26—29 that , 1 ah h — Qri E (s, @, @,, @) — A =§ | | {bs (w) + log wu — C3} a=0 / 0 a’ SE 82 tS deca2000 (79); eo tg py ele ~ {abs (w) + log uw — Cy} Jo f=} ys and generally it appears that if I(s)>0 and is not integral, — 27 Ec 1, @,, @») — Al 5 E | “ {ds (wu) + log u} Ss "ey See (80) 4 peek ( ay = Id) i {yr (w) + log wu} aH If Ws) is integral a new set of formulae is required, of which we have already worked out those which correspond to the cases s=0, 1,2. The form of the general formula for positive integral values of s may be readily inferred from these examples. DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 31 PART IIL THE CONNECTION OF THE FORMULAE OF THE PAPER WITH MR BARNES CONTOUR INTEGRALS FOR THE DOUBLE ZETA- AND GAMMA-FUNCTIONS. § 32. When the real parts of @,, w, and @ are positive, the double Zeta-function may be defined* for all values of s for which it exists by the contour integral wr (1 — | Ce (— 1th eae In ~| dae) (een Fic. 3. where (— uw) = e* 8-4), log (—u) being real when w is real and negative, and rendered uniform by a cut along the positive half of the real axis. The contour of integration is a loop from u=+20 to u=+ enclosing 0 but no other pole of the subject of integration. Suppose that the contour is separated at x» and completed by drawing a very large circle whose centre is 0. Then under certain conditions which I shall formulate shortly the value of the integral when the radius of the circle is made infinite will still be the same. If this is so it is clear that 2a a a (8, G5 Oy; Wa AFC Say ? 1 ») will be equal to — 277 times the sum of the residues for poles within the contour. Now the roots of 1—e—” are and the residue of et (— up ie! = Eo) (1 ay Ca) 2 2Qn71 . at u= 1s @, _2nart iy fer 2n7v\ n @; _ nari (- @; ) l-e ™ * Barnes, loc. cit., p. 314. 382 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND The sum of all these residues from n=-— 0 to o (exclusive of 0) will be convergent if lim 1 n=to Qnari—-2n(a—w,) zt —AU Perr rrrrr rrr rere ere eee ee ee (82), eo —¢e [oy which will be the case if IR ) and 32 {@=9)*l tave opposite signs. 1 \ @) ) But if w and w are two complex quantities =2,+7y, and 2, + ty, respectively, the Ust 1 meaning of IR (=) >0 is that A s Ye / é by U 1 & Hence it is easy to see that (82) is satisfied if @ lies within the strip bounded by (0, © @,) and (w,, @.4+@,). This is in fact obvious on a reference to Fig. 4. Fie. 4. The roots of 1—e-*" give another series of residues which is certainly convergent if a lies in the strip bounded by (0, 2 @.) and (@,, @; + % @2). Hence if a@ lies within the principal period parallelogram both series will be convergent. Moreover it is easy to see that if this is the case, and if the circular part of the contour, in its progress to infinity, always has its radius so chosen that it never passes at less than some assigned distance from any pole, the value of its contribution to the contour integral ultimately vanishes. For (neglecting the power of w) the subject of integration is 1 ett — e(a—a,) U _ 9 (A—ay) U 4 9(A—w, Wy) U Now if a=a,+%, w=a+iy, and |w) is large, |e*”| will be large or small according as 0 = Ay. Draw the line (0, ©a) and its image with respect to w=y. Then the part below or above this line is the part of the plane in which |e*") is infinite at infinity, according as a — 0. A reference to Fig. 5, in which the different angles within which the four terms in the denominator of (83) become infinite are indicated by semicircular ares, now shows that ett! — e(a—w,) U _ g(a—ay) U4 p(a—w ws) U becomes infinite, and so (83) vanishes, all along the circular part of the contour. Hence the contribution of this part ultimately vanishes. Hence _2nami ; _2nazi , G(S305.@\5 @s) ee eee tar Qi) \oa le € Qn = = 7 ate ? 2nwyrt en (i —s) ior Baas a) On Senay @s ( a a 1 aes bad | i - 1 ——e 9 5 i (84). DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 33 Now let us consider the first series. Take first the general term for which n=v is positive. In it 2v7i\*4 z vei (-=) ks 1) log ( = ) @) > where the imaginary part of the logarithm lies between + 7. Fic. 5. Similarly in the corresponding term for which n=—v is negative 2v7ri\4 = Bux ( ) ze log ( as @) where again the imaginary part of the logarithm lies between + 77. Now if », makes an angle 6, with Ow the imaginary part of log =") is (da — 0, + Qhrr)i, and as $7 —9@, lies between 0 and 7, k=0. Similarly that of log (- am) is {da — 0, + (2k +1) rh} 2 and here k=—1. Thus the terms for which n=+~vy give together = s—1 = -(s-1)(3r+6,)é p= = (s=1)(—3r+ 0) o ) _Qvw ort Qvegrt | -) l—e @, l—e », BeaEe : - vs Qv7r\t eé w i 2v(a—}ws)nt +4(s-1) mi oe -3(s—1)mi) = Maen Seen Wh 1 ) l—e @) . (2v7\* a . (2v7r(a—4o,) 7 ay (=) 1 = q sin jens ee =e 4 (s — 1) CER pac 1 1 Vou. XX. Parr I. 5 34 Mr HARDY, THE EXPRESSION OF THE DOUBLE ZETA-FUNCTION AND Thus the first term in the expression of €,(s, 4, @, @») 1s 9; (27) rid—s)S ¢ yo. (2u7r (a — $0.) : —=— (Sone os cecweiseree 85 oo! er ,sin | = +4( )a (85) where w,* has its principal value § 33. Now* ae BEES 9 (igs eee cae ee g(u—a—ta.)+ = 2 & er gic oe (86), for all values of w and a such that w—a lies in the strip bounded by (— ©@, + }@2, © w; + Fes), and (—xa,—t0,, 2a, —}o,), a condition which is evidently satisfied if a hes in the strip and wu on the axis of a, Hence oa . m\ du __ 4 =) a ibs soe (u-a) du i |e i aos) aa Chie Ge SOF I Q, I ---(87). Here it is supposed that 1 <4R (s) <2, and that us = e~8 log uw. log w having its principal value. Now eg - 2n(w— a) mi ee Se ah Jo P @ we oe ele ae Along the path of integration “=a, where « is real, and w= a,x. e182) = @)-8g1-* @, where @,~* has its principal value Hence the mtegral is 2-8 ea Qnert gp I—Se] pp — 28 (2 s—2 1 (2 2na wo se 4, ema — wre (207) (2 — 8) exp j— 0 = ae 12-5)| nik, oe _ m) du (2a 5 . a fe loc a as o,) wo ee Fhe >To 2na 2 lexp ; = 4(2— | 7 + exp a = -4(2- | nil RSb SS aco (88). cos | ae 42-5) @s Here ,* has its principal value, and a lies within the limits previously defined. If we write a—t, for a, the new a will lie in the strip bounded by the axis of , and the parallel to it through »,, and for such an a “Foun a) i) dH 9 (2) zn I low a+ a n 19 T(2—-s)s q arene (2— 8) = 7a 008 |= and so «2 a nr == =) re-)se aes Pat @, — qn Eee (= =) T(2—s)s : ae sin men a= ba) 4@,) Now since €,(s, a, @,, @:) is the sum “ two series, of aes 24 (27) 7T 1 - QS aa @,° +(2—s) |) ...(89). 44-12} (2n7r (a — ta. n4 {ye 2446-121 * Halphen, Fonctions Elliptiques, 1. 426. The notation is different DOUBLE GAMMA-FUNCTION IN TERMS OF ELLIPTIC FUNCTIONS. 35 is one, it follows that — 277 (s—1) €.(s, a, @,, @) is the sum of two series, of which one is the series which occurs in (89). The second series may be treated in the same way; and so we arrive at the formula 0 @) — Qari (s— 1) cs (s, a, @), @) = er: le (u — a) + a} = -|~ |e — a) +m HOOrei (90). Now if in (34) we suppose r+3—«=1, e=r+2, we obtain Ow, ~2ni(e+r +1) b,(e+r+2, 4,0, 0,)=][ o,(w—a) udu 0 -|~ vr, (u — a) us du { which agrees with the above formula when we put s+2=¢ and suppose r=0, so that —-1 +41 p—2! 23 ke 2h, 2p ! ke 2p! P p—3 1 Pp: A +3igip—siges 91+ Gip—aigae to + Also co pa ky ky, ae ks; 8 hay ae 7! 7! @ = aor op hihi, + arg hokst gigi hilet oy tee 2p +1! 2p+1! ape) — pt »p—-lp. pt . »p—2}. x 3ip—1igea' i, +3Tp ator heist. From the point which has been reached it is an easy step to the sought representation of the groups by means of definite integrals. Observing that = kv is the integral between | 2p limits + of the expression e 2% we obtain this function for y, as the first approximation to Vr2k +90 the law of error, since the difference between | atydz and the mean tth power of the actual locus is small. The difference is small, in the case of an even power, relatively to the value of the power whether the actual value or that given by the approximation is taken, being of the order = with respect to either. In the case of an odd power there is some difficulty in applying this test, as the odd mean powers are zero for the representative locus. The difficulty would not exist if the test of approximation were, that the ratio of any mean power, say the ¢th, to a certain even mean power, say the 2rth (preferably the second), raised to the power ¢/2r, should be approximately the same for the actual and the representative locus. The difficulty does not affect the further approximations. The second approximation y, a ought to be such that at (y,)da generally should be equal to #°*” to within a small frac- ae Stet Ay . tion— mit as it proves to be—of the true value of that power. The requisite correction 3, is afforded by putting for y%, Yo — nee equal to | a°(y,)dx. And when p is greater than 1, For when p is 1, #*), that is #®, is exactly im 1 d’y, , ky ean (oy fh ees = (2 DO 1) [2 (ub 3 G8) de = 2p +) BP) —D 5, Qp+1! , p-2! 1,4 Qp—213! *p—1! 24° na x [ ge y daz = / -D Be pie 1 : which is exactly the value for 2+” when quantities of the order of 7 relatively to k,—of the order = relatively to k,—are neglected. 6—2 44 Pror. EDGEWORTH, THE LAW OF ERROR. By parity of reasoning the second correction, forming the third approximation, is found to be 1 dty, i‘ ke 1 dy 741 dat 21313! dat” ky A general expression for the series of corrections may be found by a slight change of the expression for the generating function for mean powers, In that expression omit from the index the first term 4/,62; substitute for every other @ the symbol — = and place after the operator : 1 a é thus constituted, as operand, the function St “2k, We have then for y,, the ¢th approxi- Vor 2k Gat aye ban) Og vila)” oe x 12k mation, + 2 For the mean tth power as determined from this curve, viz. | z'y,dx, is equal, to within quantities of the tth order, to the mean ¢th power as determined by the generating function. For consider any term in the expansion of x as determined in the latter way. It is, if ¢ is even, of the form Nt a Pe Ueseicer cae 3 ee eee a+2!/ \B+2! ! An identical term is given by the expansion of the operator, namely, ., (where 2r+a(a+2)+b(8+2)+...=1); say t! Pets ie Qo nde, Grhere aa 2) ob (GD) eee t! hie es wryde=t1 Qa, In like manner if ¢ is odd the terms of the two expressions for 7 may be identified. It will be observed that any coefficient of the type 4; may be regarded as the difference between the mean ¢th power as it actually is, 2, and what it would have been if the actual frequency-locus of the compound coincided with the ¢th approximation 7%. The coefficient hy may also be regarded as the sum of coefficients of the type «, pertainmg to the different elements; where «; for any the qth element is the difference between the mean ¢th powers for that element as it actually is and what it would have been if the actual frequency-locus for the element coincided with the law of error 7;_,, involving the coefficients «, « ... K1 in the same ai as 7:4, the tth approximation to the frequency-locus for the compound quantity, involves Ky, hy, ... Keo. In considering the order of the successive groups formed by the expansion of the generating function, or the equivalent operator, account should be taken of the number of terms which go to each group and of the numerical coefficient 1/t! with which the coefficient is affected. Let p; be the proportionate increase of the tth group compared with the (¢—2)th group due to these causes—any homogeneous product of the tth degree of the ks being treated as equal to any others of that degree, and equal to “th of any homogeneous product of the Pror. EDGEWORTH, THE it.AW OF ERROR, 45 (t— 2)th degree (except k, and k,*). Then m must be sufficiently large that p;,/m should be small. For any given value of m ceteris paribus the approximation becomes slower as with the increase of ¢ p; approaches mt. With these explanations we may write the general law of error Sl ary, _ldy ke 1 dy By. 9 — sige? target aisiai ae? ~” ae 2 . ie = ; : : - where y, is put for aon. %), the successive groups being given by the coefficients of succes- / ‘0 sive powers of ¢ in the expansion of 6k, (2) 6%, (2) -¢th 7 (ZY o tsi (ae) +g (ge) tot DO hE (Fe Yo Section II, The Method originated by Professor Morgan Crofton. The fundamental propositions which are required in order to find a law of frequency which is approximative in the sense that the mean powers of the representative and the actual locus are nearly equal, are also required in order to find a law of frequency which is approximative in the more familiar sense that the ordinate of the representative locus at any point and that of the actual locus at the corresponding point should be nearly equal. Of the methods which pursue the latter quaesitum Professor Crofton especially requires the support of the propositions here regarded as fundamental. His proof of the law of error is based on a datum obtained by observing the effect which the introduction of a new element produced on the frequency-locus for the aggregate of elements. It seems to be assumed, very properly, that the sought function involves as constants some at least of the mean powers of the aggregate?. We may pass rapidly over Professor Crofton’s first step which is to obtain a partial differential equation of the first order by which he deduces that, when a new element is taken in, the mean first power of the aggregate is increased by the mean first power of that element. This inference may be obtained immediately from the fundamental theorem eV SEO 4 EM + ot En§, This being observed we may without loss of generality refer each of the elements (and accordingly the aggregate) to its respective centre of gravity. Then if y,=/f(w), is the ordinate of the frequency-locus for the aggregate before taking in a new element, and y + dy the ordinate after that operation, by a well-known principle), y + 07 =[Sd¢m (&) f (w — &) AE] * k, is of the order kym =f: § Asimple case of the multinomial law for a mean power, + On the value of p see the Appendix. as to which see the Appendix. + It is a priori evident that the actual locus can be || Morgan Crofton, loc. cit. p. 781, col. a. The principle exactly expressed by a function involving as constants as had been used by the present writer in his first publication many mean powers as there are values—multiples of Ar— on these subjects in the Philosophical Magazine, 1883, assumed by the compound quantity in the long run. Vol. xvr. p. 301. 46 Pror. EDGEWORTH, THE LAW OF ERROR. where 7 =¢,,(€) is the frequency-locus for the new element, and the square brackets indicate that the summation is to extend over the whole range of values assumed by that element. Expanding in ascending powers of (each value of) £, and neglecting powers above the second, as is found to be legitimate under the conditions specitied*, we have (since the first mean power of the element vanishes) dy =} [SE%bm (£) Az] CL di?” From the fundamental proposition that oO = E+ 204 04+ E97, it follows that [S&,, (&) A€] the mean second power of deviation for the mth element is equal to 0k + the addition to k the mean second power of deviation for the aggregate. There is thus obtained a partial differential equation of the second order dy _,@y | ame PR (1). In order to obtain a subsidiary condition wherewith to supplement this leading equation a further recourse to the theory here proposed as fundamental may be made. A subsidiary equation is (in effect) obtained by Professor Crofton from the property that if the unit according to which the axis of # is graduated is altered in any assigned ratio, there must be a corre- sponding alteration both of the ordinate expressing the frequency of the aggregate and of the mean square of deviation for the aggregation. By supposing the alteration indefinitely small he obtains a second partial differential equation, viz. (in our notation) yo 4 op 9 Se ae (2) Now the form of y which is given by the general solution of this equation, viz. 1 ( ie ) y= — —=), TE” WI where Y is an arbitrary function, may at once be presumed from the relations of the ordinate and the mean-square of deviation to the abscissa which have just been adverted to; provided that we assume—as Professor Crofton seems to have assumed—that the sought function involves only one of the mean powers as a constant, namely the second. Otherwise why should we not have in the last written general expression, instead of Vk, as the denominator of x (and Av) the fourth root of the mean fourth power; with a corresponding modification of Professor Crofton’s subsidiary partial differential equation? The justification for omitting # (and higher mean powers) is to be sought in the fundamental principle that the mean fourth (and every higher even mean power) differs from a certain function of the mean second power by a quantity which may be neglected in a first approximation§; while the mean odd powers differ from zero by a quantity which may be likewise neglected|. * See the appended note on the orders of magnitude + Identical with the k, of Section I. Where no mistake involved. can arise k has been sometimes used instead of ky. + A simple case of the multinomial law for a mean § See above, Section I. power, as to which see the Appendix, || Above, p. 42. Pror. EDGEWORTH, THE LAW OF ERROR. A7 The subsidiary condition having been established in either of the forms above written, it may be combined with the leading equation either by treating the partial differential equations as simultaneous after the procedure adopted by Professor Crofton *, or by restricting the general solution of equation (1) in the manner proposed by the present writer. It may be remarked that Professor Crofton’s procedure seems to require yet another recourse to the fundamental theory. By combining equation (1) with (2) there is obtained a first integral yx + = On from which the normal law of error is obtained, if C is zero. C is inferred to be zero because, when vanishes, dy also vanishes. But can that premiss be obtained otherwise than by the dx fundamental theory that the odd powers (above the first, which is zero by the construction here employed) are negligible in a first approximation, and accordingly that y may be regarded as involving only even powers ? The first and main part of the structure having been placed on a firm foundation, let us proceed to repair and add to that superstructure which is formed by the second and subsequent approximations. By continuing the expansion for the ordinate of the sought locus when varied by taking in a new element, as above represented, we obtain the additional term 1 z dy — 3 LSE bm (E) AE] ie" By the fundamental theory the quantity represented by the bracketed sum is 0h, the addition made to k, the mean third power of deviation for the aggregate. As the increment dh, (= «,) is evidently independent of the increment ok (=«,) we obtain for the first approximation y, the partial differential equation dy ol dy, Fee Te ee (3). The employment of this equation is restricted by the condition that /, is small relatively to yo and k,. Accordingly y may presumably be expanded in ascending powers of *, as thus: . | ay Y=Yor k, alee Sen Employing equation (3), and neglecting quantities below the order 1/Vm, we have for that 3 function of « of which 4, is the coefficient in the expansion of y, = ly Mere Ss v- L y 1 dy, = 4; — Yo, is thus — hy = a + When we go on to the third approximation, there occurs the difficulty that the fourth mean power of the superadded element cannot, like the third, be treated as an independent variable. In assigning 0k = «, = &"!, the increment of k, we have also assigned a part of &,,. What may be regarded as independent is the difference between the fourth power as it actually is and what it would be if the element were perfectly “normal,” viz. &* —3«,°. By the fundamental theory * loc. cit. t+ Cp. Phil. Mag. 1896, Vol. xut. p. 93 et seg. (by the t+ Phil. Mag. 1896, Vol. x1. p. 96, top. present writer). 48 Pror. EDGEWORTH, THE LAW OF ERROR. this quantity, «2, is the increment contributed by the new element to k, (=«* —3h*). To connect this increment with the increment of y that depends on it, there must be utilised a fresh datum, namely that the aggregate formed by taking on a new element is itself obedient to the law of error; and accordingly satisfies equation (1). Whence d 1 dé ap Y + 8Y) = 5 Ga Yt &Y) where dy is the addition made to the ordinate by taking on a new element. Therefore dy. dy AEs Ady (ad ania Gk 9 daz” ak = 2 dae” 2 dat (se) = (5az)- Now continuing to ae the variation of y due to the introduction of a new element, we have dy dk? + 6k y, + 6k +5 5m SY 7 + By, +. ae =, + SUSE Gu (E) ALITY, — ELSE Gn (E) AE SY, + gy [SE Gn (©) AE] SY where dy, is the variation of y dependent on the increment 6,, and the other symbols have the meaning already assigned. By the condition just established Bringing over the term in the latter form from the left to the right-hand side of the equation, omitting terms that have been proved to be equal on both sides, and substituting equivalent symbols in the expression for the remainder, we have 1 dys _ 5). 1 déy, 4! da "4! dat dy. by. = (E" — 3 (dk)?) (approximately). 1 dye. Whence PT ae subject to the condition that &, is small, of the order = By that condition y may presumably be expanded in ascending powers of k,; and thus, by parity of reasoning with that employed for the first correction, we obtain for (y2— %) 1 dy "4! dat +R; where R is of the nature of a constant with respect to /,—involving only k,. What & is may be seen by continuing the expansion of i in ascending powers of f;: ey a aes since by a parity of reasoning to that just now wee in the case of the condition dy _1d*y dk 2 da?’ Pror. EDGEWORTH, THE LAW OF ERROR. 49 : dy\ 1 dy, not only is (St)=- tae é; d\? 7 1 d\ but also (ae) ¥=(- 51g) With the aid of the fundamental theory the method of partial differential equations can be extended to any number of approximations. The general, the tth, correction may be ex- pressed as follows. Defining & as the mean square of deviation from the centre of gravity of the compound quantity whose law of frequency is to be represented, /, as the difference between the mean third power of deviation for that quantity as it actually is and as it would be if the first approximation y, were perfectly accurate, k, the difference between the mean fourth power as it actually is and as it would be if the second approximation y, were perfectly accurate, and so on, expand y, by Taylor's theorem as if it were a function of the variables /,0, k,62, ..., k,6¢, and write out the coefficient of @‘ in the expansion thus : dye] ,. ay |p. 7 Le |, x &. Bake + Esa Wa +... + i! Fak Ke : where after the differentiations ,, k., ... kh; are replaced by zero. Now substitute for - ats Se ee th veal sat Cal Spal i every operator 7 % operating on y;, the operator p+2i daP® operating on y,; and for every power of the former operator the corresponding power of the latter. The resulting expression is the tth correction, the difference between the (f+1)th approximation, y,, and the ¢th approxima- tion, 7. For assuming that the required expression y involves mean powers of deviation* as constants, we know by the fundamental theory that they must enter in the forms h, h,, k,, &e., - : ; 1 1 ; with orders of magnitude proportioned to 1, oe — Therefore the above-written vm m rule for the tth approximation will hold good, provided that dyq _ (—1)7 da? @) dk, q+2!dat 7” aN f a\" (—1)? d\s/(—1)! d and (II) ce) = = Sac ( as) sco Uy) where p, g, ... are integers up to ¢. Now proposition II. follows from proposition I. by parity of reasoning with that above employed+ with reference to equations (1) and (3). And it may be proved that if proposition I. holds good for A;_, and all coefficients with lower subscripts, it holds good also for ky. For continuing the expansion of the ordinate? when a new element is taken im up to the tth order of magnitude we have for that order on the ae side of the equation = s r - dt*2 (— 1)? 5 [Sm BEE] FE = (— a, BO ao * Cf. note to p. 45 above. + Above, pp. 48, 49. t+ Above, p. 46. Wiow, MOS Lenin IE 7 50 Pror. EDGEWORTH, THE LAW OF ERROR. On the left side of the equation we have oY oh, + OF aki, + 3 GE Be, ee up to terms of the ¢th order of magnitude. The terms preceding the (¢+2)nd group* are cancelled by equal terms on the right hand of the equation. The terms following the (t+ 2)nd group may be neglected (with reference to the tth correction). There remain on the left hand to be set against the term on the right which has just been written a group of terms of the type aa soo OE ria! b! dx,’dx*dk which the contribution to the variation of the ordinate y by the (¢+2)nd mean power of the new element is involved in the assumptions which have been already made with regard to > ¥, Where 2r+a(a+2)+b(@+2)...=t¢+ 2+. Beside these terms, of B earlier mean powers of the element, there is to be placed a contribution to y, which is not thus dependent on preceding assumptions, dy; as it may be called, or 0(y:—y4«), dependent on the independent portion of &,,. Now proposition I, and therefore proposition II, are supposed to hold good up to and G3) 9- (Gr a) ™ occurs in the group on the left with subscript inclusive of the equation Observing that no differential of the type ze Wiles When ol me seal h symbol ( ‘yy its equivalent (‘— ian) 3 ee nigher than ¢—2, we replace each symbo ae y q q+ 2! dat) ° an us iT tte obtain on the left hand an expression of the form &,/¢* Pere Sn +0, where E,/®) is the (¢ + 2)nd power of the mth, the newly added element, on the supposition that its frequency- locus exactly coincides with the law of error m4, a frequency-function of the same form as y%, but involving as constants the coefficients of the element «9, 4, ..., Xt. Which have been already utilised—the portion therefore of the (¢+ 2)nd power of the mth element which can no longer be regarded as independently variable. What may be regarded as variable in the mean (t+ 2)nd power is &,,+? — &,,/*, the ditference between the power as it actually is and what it would be if the approximations hitherto made were exact ; in other symbols «;, which by the fundamental theory may be regarded as 6k;. Thus bringing over from the left to the right the portion affected with &,,'"* we obtain dy, _(—1)' dy, dk, t+2! dat’ (approximately) ; * Or the tth group in the notation of p. 41 above. + It will be remembered that ky is of the order ky/m, i.e. 2s kp is of the order ~?** (above, p. 42) that is of the 1 order (— Vm pt+2 ) , or one mth of the order of Kp. Pror. EDGEWORTH, THE LAW OF ERROR. 51 which was to be proved. Therefore if proposition I. holds good up to the (¢—2)nd approxima- tion it holds good also for the tth. And it has been shown to hold good for the first and second approximations. Therefore the proposition holds good in general. Therefore in general 1 dt (t—-Yta)= a a where R consists of terms of the ¢th order of magnitude not involving k;. What those terms are is found by expanding y in powers of h, 0, k,.@, ..., k;,@*7 and selecting the coefficient of 6*, and (<1 drt Gare The series of terms thus formed are identical with that obtained in Section I. substituting for every symbol of the type ( a) " the symbol ) operating on yp. - ‘p/ 0 A variant solution of the system of partial differential equations, in the form of the real or significant portion of an expression involving the symbol 2 S , will be given in the Appendix. Section III. The Method originated by Laplace. The method originated by Laplace and developed by Poisson seems also to require for its extension, if not for its support, the propositions here regarded as fundamental. In a free version of this classical investigation let us begin with the case to which Laplace confined himself, the comparatively simple case in which all the elements have one and the same locus. Let that locus be = ¢(€), signifying that the probability of the element having a value & is ¢ (&) AE, where [Sé(&) AE]=1, the square brackets denoting summation between extreme limits. Put y (a) for [Sd (€) e¥~'A€], where, as all along, & is an integer multiple of A€, (or Az) = pAz, say. Form the mth power of x(a). The coefficient of eV —larax in (x (a))™ is the probability that the sum of the values of the elements should be equal to rAz; a probability which is equal to yAw where y is the ordinate of the locus representing the frequency of the compound quantity. Now multiply y”(a) by e V-Ware The product will consist of a sum of terms all of the form ¢e"-947«V-ly,.,. (where r assumes every possible value) ; except one which is free from the transcendental expression, viz. y:szAz. Each of the other terms may be put in the form ce Az (cos (r — t) Awa + V—1 sin (r—t) Aza) ypax- Put 8 for Axz and integrate the expression thus transformed, with respect to 8 between limits 8=0 and B=7. All the real terms of the integral vanish except 7ys,4e. An imaginary part in general subsists. Thus we obtain as equal to 7,,,Az the expression Rr Ar—| - x (B/Az) e-V-18 dp. * A symbol used to denote the useful conception of the real part of an expression (a conception used with effect by Czuber in his Beobachtungsfehler). 7—2 52 Pror. EDGEWORTH, THE LAW OF ERROR. Substituting # for tAx, omitting Aw from both sides of the equation, and changing the inde- pendent variable from 8 to a, we have ye m/Ax Yx = NT | x™ (a) e~¥—laxda, The path from this point is easiest when* we take the frequency-function of the element to be symmetrical. The expression which has been found for y may then be evaluated by expanding y(a) in ascending mean powers of the element 6 ee 1 —,.» 1 xX (a) = Is @ 5 oP E+ 41 até +... + (V—1)” Ip! are?) ae| 2 where & assumes all possible values between its extremities. Whence 1 x(a) =1— 5 ve + Fate — To determine {y (a)}” take the logarithm of x (a). eral =T) a2 Log X (2) =O — 55 Oly + Fy athe — «ee + (V1) a? ap 1 23 where x, 2, --. have the signification attached to them in our first section. Accordingly 1 1 1 log {y (a)}" =m log y (a) = — 5 atk, + rn hoot — ... +(V— 1)? Dp he where k,, ky, ..., having the signification before assigned to them, form a series descending in magnitude. Thus a/A% TA pg ha — y=H+ | Py Ben Vcc am Tin 0 1 pase Vie tp ——— . R= e 2 tai!" (cos aw — V— 1 sin ax) da ; Tso 1 priae i 2p24 + 7, = e2 ° 4!” “cosazrda; TsO which may be expanded in powers of small quantities ie — hark 1 ~ od 1 n ab 1 =) 8 uf i : alk e-3 Lt phat = bah + 5 (a arb eacon ae as the rather as each of the definite integrals thus presented is not only finite, but also is made up m/AZ 4 of finite elements. The first approximation is therefore — [ e- #0 cos aada; where Az is at To most of the order 1//m+ and may be indefinitely small. The expression therefore differs by a very small quantity from ek, oc | e732 cos aada = —— Jo Vr2Qk the normal law of error. * The facilitation proper to the simpler case is intro- element is Ar which with reference to magnitudes that duced by Laplace at an earlier stage. are equated to unity is of the order 1jV m. See below, + E.g. in the case of the Binomial the range of the Section V. and Appendix. Pror. EDGEWORTH, THE LAW OF ERROR. 53 FA See Pape m/c A nearer approximation is made by taking in the term = ke / ate-*ko cos avda. Now Lelo this may be identified with 1 n/Ax 4 ; 1 ds = in — e-42k eos ae) ae 5 nh, daa’ cos axda aia Yo} where y is the above written expression for the function which forms the first approximation. In like manner any other term of the expansion of y in the series of definite integrals may be identified with a term in the series obtained by the method of Section I. The general term for the 2pth correction by the present method is — kea® keg? 1 1 Sree wee Wy bi @+2) (842 alk dab? i*o cos axda*, where 2p=ar+bB+..., P=a(a+2)+b(8+2)+...; which is identical with a term obtained by the earlier method since or —— > COS ar = (V— 1)" a" cos aw. da When the odd powers of the element do not vanish y(a) is to be written L iW a wie (14+ V=1 age — Soe" — 2 V1 EM +..); where £" vanishes, the elements being referred each to its centre of gravity as origin. Thus 1 1 ee 1 mW, See) _— 1 1GEOE log x (a) =— 5 ae — 50° nN aa E% +a! ( jf —5 5 Ey) +4 — Ia (— FEO +5 aE) fi 1 —- er == 1 =— 5A —_ eV —lat lla mt (Vl a et... Which differs from the mdex in the generating function (or the corresponding operator) pertain- d ot x) there employed there is now substituted V—la. The expression for m log y (a) = log (x (a))” is similarly related to the generating function and operator pertaining to the compound. Thus ing to an element, given in Section I. only in this, that for the @ (or — 1 1 = — 1 -5a%y-s a V -1k,+...(V 1) ata hee a ee y=Rfe 2 a” rely **"e™2 (eos aw —\V — 1 sin ax) da. Expanding in powers of the small quantities /,, k., etc., and selecting the real terms we obtain two sets of terms of the respective types (Vv —1yP =! | a2P e~tka* cos anda ) ( and — (Vaaypeet i | o2P +1 etka? sin aada, 0 where / and /’ are products of ks (divided by factorials). * w being substituted for r/Aé as before. 54 Pror. EDGEWORTH, THE LAW OF ERROR. Every term in the expression obtained before, by expansion of the form 21\ dz ke d\t+2 Gace ie corresponds to a term in one or other of these two series, as will be found by observing that when ¢ is even t —— (-£) cos ax = (V — 1)tat cos aa; and when ¢ is odd t ae (- 5) cos ax = —(V — 1) at sin aa*. So far it has been assumed with Laplace that each element has the same frequency- function. But the reasoning is not affected when we relax this condition to the extent permissible in the typical case ; substituting for (x (a))” the product x(a) x2 (4)... %m(«), Where each of the ys pertains to a different function ; and making other slight changes. The reason which legitimates the substitution of e—*° for y,(@), %2(@)-.» %m(@) as part of the definite integral (with respect to a) which is equivalent to the required function of « has not in general been clearly stated by the mathematicians who have treated this subject. Laplace indeed (Théorie Analytique des Probabilités, Book tv. Art. 22, p. 336, ed. 1847), with reference to the case in which all the frequency-curves have the same locus, has expressed the true reason, namely that the logarithm of the above-written expression in a may be expanded in ascending powers of a of which the coefficients form a descending series of magnitudes. So too Professor Czuber, with reference to the case of symmetrical elements (Theorie der Beobachtungsfehler, p. 91). Yet later, referring to Ellis’ hesitation about the Method of Least Squares, Professor Czuber argues (p. 271, Art. 117) that there is a “ difficult point” not yet cleared upf. Section LV. A variant Method. The preceding results may be confirmed by a method of proof based partly on conditions which have been utilised by Professor Crofton, and partly on a fresh condition which will be utilised by means of Laplace’s analysis. The fresh condition is that the sought frequency- function must be reproductive, in the sense that if two or more independently fluctuating quantities A, B,... assume different values with a frequency designated by a member of the family represented by the sought function, then Q a quantity formed by adding together each pair (triplet, etc.) of concurrent values presented by A, B,... will also assume different values with a frequency designated by a member of the sought family. For as A is—or may be * Cp. Czuber, Theorie der Beobachtungsfehler, p. 90, on + See appended note on the ’Amépia of Ellis and the successive differentiations of the integral. Czuber. Pror. EDGEWORTH, THE LAW OF ERROR. 55 replaced by—an aggregate of a great number, say m,, elements of the typical sort, and likewise B is equivalent to the aggregate of m, other elements, and so on; Q being the aggregate of m,+m,+ ete. elements of the typical sort must fulfil the law of error. Let us at first limit the enquiry by supposing that A, B,... each range under the same member of the family of frequeney-curves—or more generally loci—which have the property in question; and that this common member is symmetrical. Let the common frequency-locus be y=f(z). It is required to find the form of f such that, if A, B,... ete. each range under a locus of that form, then Q the aggregate of A, B,... shall range under a locus of 1 x the form heer dl (=). By Laplace’s analysis if x (2) = [Sf (x) Az cos aa], then as the ordinate of the frequency-locus for the aggregate we have y==| (x (a))™ cos axda; Tso where m is the number of the components, not in general (in the present enquiry) a large number. Put e”@ for y(a). Then the condition of reproduction will be satisfied if 1 ~ —]| e™@ cos arda Tso can be put into the form LS ies DL = S|) e) cos a ~ da. CTW Jo Put a/e= 8; and the last written integral becomes z i e¥ 8) cos BxdB. TJo Whence it appears that the condition will be satisfied if y(c8)= my (8); where c is a constant depending on m and y(§) is a function not involving m. Now the general solution* of the functional equation (ca) = mp (a) is of the form a’ (a, + a, cos 27rt log a/log m+ a, cos 4rrt log a/log m+... +b, sin 27rt log a/log m +b, sin 4zrt log a/log m + ...). But in the case with which we have to do the coefficients a,... b,... must all vanish as m does not enter into (8). The required function is therefore of the form ee) il =~ | é cos ada. 7 Jo It is necessary that the function should be of this form in order that it should be reproductive when the components A, B,... have the same frequency curve; and it is evidently sufficient that the function should be of this form in order that it should be reproductive when the components being of different magnitude belong to different members of the reproductive family. * The solution given by Boole, Finite Differences, Ch. x1. Art. 5; the symbols being altered. 56 Pror. EDGEWORTH, THE LAW OF ERROR. The general form which has been obtained may be restricted by a condition which has been utilised in Professor Crofton’s method, gs that any member of the sought family must be approximately at least of the form —— pas ( al: where & is the mean square Y of deviation from the centre of gravity, and q is a constant such that | Bux ip (= =) de= k. —2 qvk qvk This form may be combined with the recently found form - | e“ cos axda by putting « = 8/qvk, 0 so that the latter expression becomes lal: * gai (qVbit , PX 9 } qvew so OF ee When a fluctuating quantity having this frequency-locus is compounded with m—1 others of the same species, the frequency-locus for the compound is = 3 = erm’) (qk) cos Be/qvkdB; g 7 1 or, putting B = y/m*, eariaB* eos —Y_ dy, mut “= gk an mtqn k ls But by the proposition just now adduced the composite function is of the form wh ale ( Tz) QVK \QVK/ where K is the mean square of deviation for the compound, and Q is a constant such that [ ae Y*| =K. As W differs from w only in respect of constants, @ is evidently equal tog. Also by a fundamental proposition K=mk. Accordingly comparing the identical forms of the composite locus, we have mt identical with /mk, whence t=2. The sought function is of the form e*87/9"k cos ashe op. /0 Veh 2h ale In order to secure a finite form a must evidently be negative. And thus by familiar transformations we reach the normal law of error. When we start without limiting the enquiry by the assumption that the sought form is symmetrical it is proper to put via Is; f(< ) Anat =e, and to multiply [y(a)]™ by e-Y—™. By parity of reasoning it may thus be found that the required function is of the form «o Z i g(G+V=) a? ¢-V=aaxiaVE da. 0 T Pror. EDGEWORTH, THE LAW OF ERROR. 57 This is no doubt a reproductive form, but it is one which being imaginary does not concern us, or only concerns us in respect of its real part. But that real part, which may be written L (2 ; Le - : : : : ; is | e- #2 cos (Hat — a da (as G@ is evidently negative), is not reproductive except indeed 0 ay when H vanishes*; in which case we obtain a function of the same form as before (by putting g?= 2k). Without putting symmetry in the premisses we find it in the conclusion. Some varieties of proof by which the datum peculiar to this section is combined with the principal partial differential equation of Section I]. may be relegated to an appendix. It need only be added here that just as that leading differential equation is satisfied by other forms beside the normal, provided that the coefficients which enter into those forms do not exceed certain orders of magnitude, so the condition of reproductivity is fulfilled by those other forms under the same restrictions. In the expression found for a reproductive law of error, viz. z 5 ei cos anda, if we add to the index any term affected with a! or generally a’, where p is other than 2, the character of reproductivity will in general be destroyed. But if the coefficient of a* is of the order —2 relatively to k&, and generally the coefficient of a” (p integral) is of the order —(p— 2), then we may expand in such wise that the resulting series of approximations are of nearly the same form for a single component (A or B ...) and for the aggregate of several such quantities (Q). Section V. Verifications. Before extending to more complicated cases the theory which has been proved for the typical case it may be well to test the validity of the methods employed by showing that they lead to correct results in some simpler cases, which admit of being treated by more elementary methods. Such a test case is afforded by the Binomial locus, considered as a particular case of the typical circumstances, the case in which all the elements have the same frequency-locus, viz. two points separated by an interval Aw representing two values which are enjoyed by any element with respective frequency p and q. If the centre of gravity is taken as the origin, the abscisse of these points are —qgAwz and + pAz; the corresponding ordinates being p/Az and g/Az. It is proposed to compare the approximate frequency with which the sum of m such elements will enjoy an assigned value of , as determined on the one hand by the simpler analysis of which the case admits, and on the other hand by the more general methods investigated in the preceding sections. Let us begin with the case in which mp is an integer, and accordingly « the sum of m elements measured from the centre of gravity is an integer multiple of Az. - * Or at least is very small, in which case as explained in the next paragraph the curve will be in a sense repro- ductive. Vou. XX. Part I. 8 58 Pror. EDGEWORTH, THE LAW OF ERROR. According to the simpler methods the approximate law of frequency is deduced from the general expression for a term in the expansion of (p+q)”, viz. SR NTR dite Kes DG Le t= 11)) where 7 is the excess of the number of elements that enjoy the upper of the two possible values, viz. + pAwx, of which the probability is g, the number over and above mq the most probable number (out of m trials). By a well-known approximation employed by Laplace and others this expression may be represented as the product of three factors P x S x 7, of which the product of the first two PxS is formed by putting m!=N2Qrm mme-™ E + ae + =| ; 12m with like expressions for the factorials mp —r!, mg+7r!, and multiplying the portions of these expressions which are outside the brackets on each other and on p™?-"q2"; while 7’ is formed from the tail, so to speak, of Stirling’s formula, by multiplying together the parts of the expressions for the three factorials, which are within the brackets. The expression PST represents the probability that at any particular trial the sum of elements will be exactly rAwz, the proportionate frequency with which in the long run that value will be enjoyed: that is in our notation y,s,Aa, or y,Azv. FP is a first approximation to this guaesitum. It is found to be : : 92 ——. e@ 2mpq 5 Vor2mpq where 7, or more exactly pq, is of (ie. #) the order Vm. Put « for rAx and c for V2mpq Az, a quantity of the same order as those values of # up to which the approximation is ; pA Tae ; : available. We have then for y,Az the expression ——e <, the normal law of error; in which are the constant ¢ corresponds to V2 x the constant k (or k,), employed in former sections. When the same substitutions are made in S and 7, it is found that they respectively assume the forms e“t4:+-. and (1+ B,+B,...), where A,, A,..., and likewise B,, B,..., form a series descending in order of magnitudes, for values of « not exceeding the order of ¢. Accordingly the groups forming the successive corrections of the first approximation are P(A, +B), PGA? +A, + A,B, + B,), P (5, A+ A.A, + As + A.B, + A,B, +B, + BB,+ 5 Be). where the As and Bs have values found by Laplace and his successors, or to be found by continuing their work. Thus it has been found that Be 9 (Be ats 2mpq ( 3 Saal 2 which becomes, when « is substituted for rAw and ¢ for V2mpgq Aa, 9-p (e_2 =) ; V 2mpq le 3 (; : an expression which may be simplified by putting x for a/c. Pror. EDGEWORTH, THE LAW OF ERROR. 59 Again, A, has been found to be (in this notation) 1—2pq .,_11-3pq,, = x 2mpq 3s mpg Also B,=0, B,=—(1 — pq)/12mpq. We have thus for the first correction p-4LZ_ (x — 2°), V2mpq The second correction is found as follows: 1,2 1-49 ,._1-4pg1 .,, 1-499, x + 2) ~"ampq * mpg 3 Impq _1—2pq se 1—3pq1,, 2mpq mpg 3 ” ce se 12mpq’ AGB —O; Second correction =P (- ee ce Le pm Sy ci 2 x*) : l2mpq = 4mpq dmpq Impq These expressions for the first and second corrections are now to be compared with those which are obtained by the methods set forth in the preceding sections. According to those methods the first correction is 1, dy, = age: where Oya e, d ld or if we put x for = (4-= a = he why © a =-5)h5%, if k=k/e For the second correction, using similar notation, we have Teedtyne ki drys ai Gxt * 2 31S! dx To effect these corrections, we have =m IZ Slales hy, SES £,, in our former notation, = m(Azy pq(p—4); p=1 aaa / or ay (P=9 c=NV2mpqAz; c= (2mpq)? Aa*; ~ 2 2mpq PY Likewise Bs asad ey _ (ss af 2) \2) /pA¥* — See ee Cun “tea * The values of k, and k, might also be constructed from moments in Professor Karl Pearson’s notation (Trans. Roy. the compound curve, being respectively u, Az, (u,—3u") AE* Soe. 1895, A. p. 347). where us, “3, #, are put for the second, third and fourth 8—2 60 Pror. EDGEWORTH, THE LAW OF ERROR. The derivees of y, which will be required in the sequel are as follows : ds Ag Yo = (12x — 8x*) yo, ds . 7 atte (12 — 48x? + 16x*) y,, dé xa Yo = (— 120x + 160x* — 32x") yo, = | Yo = (— 120 + 720x? — 480x! + 64x°) yy. Utilising these data we obtain for the first correction, y,;—y), 1 ad iN ey dD 2% 131 dx ~ NImpq (x 3c ) ye For the second correction 1 dy, _3-18pq_2(1 — 6p9) ta Se fen Smpq 4mpq 12mpq ® he _1_d'y,_-(—20pq)__ 5(1—4pq) .,_ 1014p), , 11-499 ,, 2!(8!? dx® = 24mpq 4mpq 12mpq 9 mpq — Lg 8 8pg 4g eng ee TOE = 12mpq 4mpq =e 3mpq A,B, eee ee = 0, BABS ascacs sf BL ees. a JB) sagndo * (p-g)A—pq) Jélilety ondase = - Giese Pind iy) aneabes = "G8? * Trans, Roy. Soc. 1895, p. 347. + Theorie der Beobachtungsfehler, p. 85. Pror. EDGEWORTH, THE LAW OF ERROR. 61 Coefficient of x in ys—y2= ea On the other hand we have by the general method for the third correction 1 dy BE 1 ay, ky ay. 55! da *3! 4! da’ 3!(38!" dx’ where *, and k, have the significations already assigned to them; and =he/ => (E" — 10E%E)/(2pgm)! (Aa*) = mpq (p — g) (1 — 12pq) (Ax)*/2pqm x (2pqm)? (Ax) =(p—q) (1 — 12pq)/20%*. The additional derivees of y, which are required for this calculation are as follows: —k. a = 1680x — 3360x°, Dy, _ + ax =— 30240x+.... We have thus for the first term of y;—y. ye dy, (p— n=12P9) 351dx> =f teid are Tyo __ (p—q) (1 — 6pq) 35, 314! dx? 40° ae Card By ee as es (3!) dx? 403 pe which agrees with the result obtained by the more direct method. So far we have supposed mp to be an integer. The modification required when this is not the case is thus expressed by Todhunter. Put for the number of times that the “event” corresponding to the left or lower value of one of our elements is most likely to occur, out of a total number of trials m, not now mp but mp+z, where z is a proper fraction. The most probable number for the non-occurrence of the event is accordingly mg—z. These corrected values are to be substituted for mp and mg respectively in the expression for the general term of the Binomial; in which mg+7 becomes now mg+(r—2), mp—r becomes mp—(r—z). The expression thus transformed gives the probability that the number of times the event occurs will be less by the integer 7 than the most probable number, now mp+z. Now this modification is in accordance with that which is required by the general formula in order to answer the same question. The general formula, when the centre of gravity is taken as the origin for each of the elements, and so for the compound, gives the probability that any assigned number out of the m elements should have the upper— * The coefficient k; may also be calculated from the the binomial (Phil. Mag. 1886, Vol. xxi. p. 30), or by the compound locus; for which the mean fifth power may be more general method pointed out by Prof. Karl Pearson obtained by a continuation of a method employed by the (Trans. Roy. Sor. 1895, A. p. 346). present writer to find the mean second and third power of 62 Pror. EDGEWORTH, THE LAW OF ERROR. and likewise the lower—value in terms of w, the distance of the point corresponding to the assigned number, from G the centre of gravity for the aggregate. When mp is not an integer, that distance is not an integer multiple of Aw, If it is required to express the said probability in terms of that integer multiple of Aw by which the point designated « is distant from the point corresponding to the most probable number of events, say the point G’, distant by less than Az from G, then we must transform the expression for the probability in question from the origin G@ to the origin G’—on the left, at a distance zAw from G, corresponding to Todhunter’s supposition that the most probable number of events is now mp+z. Let the abscissa measured from G@’ be 2’. Put «=«'—zAw; corresponding to the change of 7 in Todhunter’s expression. The modification of the second approximation obtained by him is thus equally obtained by the general formula. The simpler analysis proper to the Binomial may be applied to verify the general theory in a less degraded case, in fact the most general case to which Laplace extended his proof, the case in which all the elements have the same law of frequency. Consider first the particular law of frequency with which Laplace introduces his general proof, the case in which the law of frequency common to all the elements is the next simplest after the binomial, viz. such that over a finite range of the axis x say from 0 to 2a, one value of each element occurs as often as another in the long run. Let these values occur at intervals equal to Aw; and let there be v such intervals in the range of the element ; so that y=2a/Az. To begin with, let v be a power of 2, say 2. Then the frequency with which an aggregate of m such elements taken at random enjoys any assigned value, may be determined by the following construction. Suppose a body of which the dimensions may be neglected, a grain or particle* moving along a line from zero in a positive direction, to take 7 steps of the following description. The first step is either zero or a, the alternatives being equally probable. Likewise with equal probability, the second step is either zero or $a ; the third step is either zero or }a; and so on up to the rth step which is either zero 0 or a/27 = 2a/27 =2a/v=Azx. Let m grains be started at the origin O on this course, and distribute themselves over the range OA, =2a, by taking 7 steps of the kind described. As each step is independent of the others, it comes to the same whether we aggregate 56 é 1 m parcels, each parcel comprising the results of 7 steps, of respectively a, gee + 24, or z parcels each comprising m steps, the first parcel comprising only steps of range a, the second parcel only steps of range $a, and so on. But each of the latter parcels corresponds to an even binomial, with ranges respectively a, $a, }a,... Aw. Therefore by the preceding para- graphs the aggregate for each of these 7 parcels fulfils the law of error; the coefficient & for the successive aggregates being ta’, a2... } (Aw). Now when two or more quantities each fulfil the normal law of error, their sum also fulfils that law; the coefficient & for the sum being the sum of the corresponding coefficients pertaining to each of the componentst. Thus the quantity is formed by the aggregation of all the steps pertaining to any one particle * The construction might be illustrated by a modifica-- another according to the formula given on p. 45 above: in tion of the error machine devised by Mr Francis Galton. general the proposition follows from the postulate that the + In the present connexion this proposition had better _law of error is reproductive. be proved by actual superposition of one normal curve upon Pror. EDGEWORTH, THE LAW OF ERROR. 63 fluctuating according to a regular* law of error of which the standard deviation squared is Lot Ba tei aol ae ees Sa a% (14g +7g+~-gs)= when y is indefinitely great, Az being indefinitely small; a result which agrees with the result which Laplace obtains for the mean square of error on that supposition, viz., When Az is supposed to decrease, y to increase indefinitely, v may be treated as an integer- power of 2 without loss of generality. The method of changing the order in the accumulation of parcels may be employed to obtain a more general verification. Let the range of each element be as before vAz. Let there be taken steps now each of the same range 0 or Az, these alternatives occurring with different probability for each step, for the first step the probability of 0 being 0,, that of Az being q: (pi: + %:=1), for the second step the probability of those alternatives being respectively p, and gq, and so on. The distribution of a large number of grains each of which has taken y steps of the kind described, is set forth in the following scheme: Stages on the course 0, Ag, 2A... (v —1) Az, 2a. Probability of a particle coming Se ein aks sia i PrPr---Pns SqiPs--- Pn» SQidaPs--»Pn--» SpreGs---Gn» 192+» In- Given distribution of frequency Pas Ue l> P. Qees P, p=I> ‘BS The last line is intended to represent the frequency with which an element enjoys each particular value 0, Az, 2Az...2a, respectively P,, P,... where 2:P=1. It appears therefore that the supposed given frequency distribution of an element can be replaced by a system of vy steps such as have been described, provided that each of the probabilities in the second row is equal to the corresponding proportionate frequency in the third row. As the sum of the fractions in the second as well as in the third row is equal to unity, the (v + 1) conditions involve only v equations for the v quantities p,, p....~n-. The equivalence between the constructed system of steps and the given multinomial distribution of an element will be attained if the following v equations are satisfied. SQiPaPs --- DP» _P; SIA Oye) Pe 1 =—, 2) == =... (1) Pipe «++ Pr Py (2) Pipe -++ Pa PS Put , for q/p,, @: for q./p2, and so on; also A, for P,/P,, A. for P./P,, and so on. Then the values of w,, w:, etc., w,, are given as the roots of the equation o” — Ayo’ + ete. + (—1) A, =0. As A,, A,, ete. are essentially positive, the equation can have no negative roots. * “Regular” is here used in a sense wider than + The difference between the arrangements correspond- **normal,” to denote the extended law of error which has __ ing (1) to the supposition that 2a=27 Az, or (2) to the sup- just been verified for the Binomial. position that 2a2=27—1Az may be neglected. 64 Pror. EDGEWORTH, THE LAW OF ERROR. First, let all the values of » be real and therefore positive. Then the aggregate of which the frequency-locus is required may be considered as made up of v parcels, each parcel con- sisting of m binomial elements with range Az, for all the elements in all the parcels, and frequency the same for each element in any one parcel, but in general different for different parcels. By the preceding paragraphs the regular* law of error forms the approxi- mate frequency-locus for each parcel and therefore for their sum. The mean square of deviation from the centre of gravity for the aggregate is the sum of the mean squares pertaining to the elements, viz., = = S mp,q,A2. aa Next let some of the roots of the equation in » be imaginary. A pair of imaginary roots a+*V—1f corresponds to two successive binomial steps each of range Av with the imaginary frequency-distribution (:), for the first step, * 1 as W=0 p" ie 1 Tle 6) ae The result of two such steps taken in succession by a great number of particles may be considered as resulting in a distribution of this sort. The particles will be massed at the three points 0, Az, 2Av with the respective real proportionate frequencies 1 ; 2a {(l+ayrt+h}+28°? +a a+ 8? TAP emeaeyae ee tea a aye+ ae 4 + = = eo In other words the quadratic factor of the equation which yields two imaginary roots corresponds to a trinomial step, with three alternatives 0, Av, 2Az, occurring with respective frequency, 1—P— Q, P, Q; P and Q—unlike the p and q from which they are derived—being real fractions. pr n=l—-p, Ps gz = 1 — pr. Following the analogy of the method proper to the binomial, let us investigate the probability that the sum of m such trinomials should differ from the most probable sum by just rAw. The most probable sum is equal to m times the mean value of the (trinomial) element, viz. m {0 x (1—P—Q)+ Aw x P + 2Ac x Q} = mAz (P + 2Q); which may be considered as an integer multiple of Az, with a loss of generality that is very easily repaired. Now the frequency with which out of the m steps (mP +s) are of length Aw, while concurrently (mQ+#) are of length 2Az is, by well-known principles, m! 2 : mP+s (ymQ+t — P— Q)ymra-P-OQ-s+ Prana. =a) =s=ns Q (=a) ; Proceeding as in the simpler case with the aid of Stirling’s law we find as a first approximation to the above expression 1 exp.(-} 4 if 11 (42 ). m2rVPVQV1—P-Q ~2mP 2mQ 2mA—P-Q) * So far at least as verified in this section, i.e. up to the fourth approximation. Pror. EDGEWORTH, THE LAW OF ERROR. 65 We have now to introduce the condition s+ 2¢=7. When the resulting value of ¢, viz. 4 (r —s), is substituted in the index of the above expression, the index becomes Meee eee Le ORS | 2mP~ 2 4mQ ~ 24m(1—P—Q)]° Put rAz as before =a and also sAw=y, and we find for the required frequency an expression in terms of « and y, where y may have any value between extreme limits. To obtain an expression free from y it is proper to integrate the expression in terms of « and y, with respect (y—azyP aw ee + oR with corresponding changes in the form of the constant outside the exponential. Put y—ax=y'; and integrating with respect to y’ between + 0 and —2 we obtain for the first to y between extreme limits. For this purpose rearrange the index in the form E : ‘ 1 approximation to the required frequency Tork exp. — 27/2k; where 2rk k=m |p (1—p) + 4q (1 — q) — 4p9q} Aa”, which is, as it ought to be, equal to m times the mean square of error for the trinomial element. Thus for every pair of imaginary roots in the equation for w a sum of m trinomial elements fluctuating according to the normal law of error is to be added to the sums of binomial elements corresponding to the real roots. Accordingly the sum of m elements whatever their laws of frequency (provided they are of the typical kind) fluctuates according to that law of error. The verification may be extended to the second, third, and further approximations for trinomial elements by an extension of the method which has been applied in this section to binomial elements. The reasoning employed in the typical case having been thus verified, we proceed with the more confidence to extend that reasoning to the general case. Vou. XX. Parr I. 9 III. On Relations among Perpetuants. By A. Youne. [Received August 2, 1904.] ANY covariant type of a system of quantics of infinite order can be expressed in terms of covariants of the form (ayas)™ (a,a3)" soo a3)8*, ee icticrioce > (i) or of the form (a,a2)™ (CAR (a3_,a3)*8-1, woaindahivedeennchneseOeer ote teeeeene (11) the sequence of the letters being fixed beforehand. All such forms are linearly independent, for it is evident that no linear algebraical relation can exist between symbolical products of the form (i) when the sequence of letters is fixed. The conditions for irreducibility of either (1) or (ii) are RS OREN Wye Gas cence hp, Again if all the letters are interchangeable, the conditions of irreducibility become AWa=1+& Aga = 24 Got See er ey AQ = 2534648 4+...4+8 My = DPE 4 2(E + Et + Esa) + Gi where the &’s are positive integers or zeros}. It has been pointed out that this result, which was proved originally for perpetuants belonging to a single quantic (in which case & must be even), also gives the conditions for perpetuant types when these are expressed in terms of products of either of the forms (i) and (ii), the sequence of the letters not being fixed§. Ibid. p. 319. Grace and Young, Algebra of Invariants, p. 379. * In writing down symbolical products we shall omit factors of the form a,,., ay 2n? * + Grace, Proc. Lond. Math. Soc., Vol. xxxv., p. 107. § Mr YOUNG, ON RELATIONS AMONG PERPETUANTS. 67 The exact number of perpetuant types of degree 6 and weight w is known to be heer | 6—2 , But when the sequence of the letters is not fixed it will be found that the conditions (iii) give too many perpetuant types; it is the first object of this paper to determine what are the relations among these forms. It will be convenient to make use of the notation of the theory of substitutions; to avoid confusion symbols which refer to substitutions are printed in Roman type. The symbol {ab...k} is used to denote the sum of the substitutions of the symmetric group of the letters a, b,... k. The symbol {ab...k}’ denotes the sum of the substitutions of the alternating group of the letters a, b,...k minus the sum of the substitutions of these letters which do not belong to the alternating group. In the last part of the paper the reducibility of certain transvectants is deduced from the results obtained. 1. Consider a perpetuant of degree 6 P= (a,a,)" (a,a3)™ Bee (a45-443)81. All perpetuants such as P will be supposed arranged in order according to the indices of the different factors, the sequence of the letters not being fixed. Thus if Q= (b,b.)" (bobs) tee (bs1bs)"8=1 where 0,, b,,... bs are the letters a, a, ...a@s arranged in some order; then Q will precede P provided that the first of the differences Ha— Ay, Ho—Aey «oe Mga Asa which does not vanish is positive. If all these differences are zero, P and Q belong to the same set, and take the same position in our arrangement. To express that the sum of certain forms like P is equal to a linear function of perpetuants which precede them in the chosen arrangement and of products of forms of lower degree, it will be convenient to write SPs The symbol & is throughout used in this sense. Thus unless the indices 2 satisfy the conditions (ill) we have P=. 2. Covariants of degree three. All perpetuants of degree three can be expressed in terms of those of the form (@,dz)* (Aods)*, AL Zu. 9—2 68 Mr YOUNG, ON RELATIONS AMONG PERPETUANTS. If X= 2p, (dy o)** (Ay dy)* — (dys) (Ade) = RB. If A=2y +1, (Gy G,)#+ (A,3)* + (da s)#* (3d, )* + (Gigh)***? (a, a)" = R. These facts are well known. They may be deduced at once from Stroh’s series*. The relations may be written (dy dy) (AaQs)* = 4 {ayayas} (a,a)* (doa3)" + RB, {a,ayas}’ (a, ay) (dod3)" = Re. When X>2u4+1, we have three independent forms (as) (da). Hence the number of perpetuants of degree three and weight w=3h+2 is 3k; for we may take w=1, 2,...%. The number of perpetuants of weight w=3k+1 is 3(k-—1)+2=3k-—1. And the number when w=3k is 3(k—1)+1=3k—-2. In every case the number is w—2; and this is known otherwise to be the exact number of perpetuants of degree three and weight w. Hence there can be no relations between these perpetuants other than those just enumerated. 3. Let j= (aa.)™ (aya) ine (Ca be a perpetuant of degree 6, whose indices 2 satisfy the conditions (111): we proceed to prove that if &.=0(r>2), then {a,aya}’ P= R. Let the symbol a refer to the perpetuant (a,a,)™ (dotts)™? 500 (Ap—2Qy)" when considered as a single binary form of infinite order, Then (aa,)"> (ypQyar)” non (eipea a3) 8 — Phe Now A, = 25 ""+2,, since &,.,=0, hence by Stroh’s series Ar /Nyy tA,y\ (2A, — 0% = ‘ : x ( “a im r ‘ (aap) (Gp Gry)! 0 BPP U i aN 2-7 ; ‘ NS a ot Np Ny 2 —l-v7 Nee A eee.) ae (Getty) 2 * (Gps) 7=0 ‘ r dh, At (Apa tA) (Ay #297 = Lt Nate ae +(=)” = ( : Pe ‘) ( fa ee 1 ) (Giese EVES (Cues). = 2 os (Gp4yy42)"" (Ap42Qy43) nee (a5143)81 =i seweticcctiserenenceeeneen (iv) But (GpGeqn) 2 (Ary)? (Gyn Oy.) +o (4 ais) 1 , rx r = (yp Ay 4)? (pay)? (AG 42)... (Ag4as) 2+ B; and when p< 2°" the perpetuant on the right-hand side can be expressed in terms * Math. Ann,, Bd. 31; Algebra of Invariants, p. 64. Mr YOUNG, ON RELATIONS AMONG PERPETUANTS. 69 of forms which contain a greater number of factors involving a,, @,4,, @ only. Thus we see that all the terms of the second sum in (iv) may be included in the symbol R; in fact this relation becomes ne x nN Ne Xe ry {— (aay) "7 (AyQrix) * +(—) * (Gryra) 7 (a,) "} (Grp Gyre)... (ag_443) 8 = lit, Nee a ny rd Whence fa, aeia}’ (Atty) 7 (Ar Ors) ” (Aria Grz2) 7... (€snds) A= RK; and therefore ca J Je 4. When £=0, we have A, =2A).. In § 2 we saw that {aB}’ (a, a) (dsa3)* = R where a, 8 are any two of the letters a,, ds, a3. Hence also {ap} (a,a.)™ (asa;)” (a,a,) ee (oesary =R; for (aa, (asa,)” (a,a,)™* see (CA — (a, a;)"» (asa) ( aya) ao (ahi as)8—1 = FR: Therefore when £ =0 {aB\’ P=R, where a, 8 are any two of G, d, ds. Similarly from the fact that {9,583} (@,G2)2*2 (a,a3)* = R, we deduce that if &=1 ener: 2 Ie The following relations have been obtained: (a) &.=0, (r>2), {aap} P=. (b) £,=0, {a,a}’P=R, {a,a,}’P=R&. (c) &=1, {aaa,}’ P=f. (d) & even, {a,a.}’ P= R, &, odd, {a,a,.} P= R. It remains to shew that there are no more relations. 5. Assuming that the relations just enumerated are all that exist between the forms which satisfy the conditions (iii), we proceed to prove that the number of these _— 26-1 = forms which are linearly independent is i Z ries a 25 w being the weight and 6 the degree. Let Dy = 22 i, Ng = 22 E ps, -- Ng = Ll fs =1- The conditions (iil) become Mot Mat fs €Ms—1, ba $ 2po together with the fact that the w’s are positive integers. 70 Mr YOUNG, ON RELATIONS AMONG PERPETUANTS. Consider first those forms for which bra = O = Mypie = eee = Moy and py, fe,.--#, are all different from zero. Let P= (a,as)" (a,a3)" meol( ey a3) be one of these forms; then by § 3 {ab}’ P = R, when a, 6 are any two of the letters G12, Aris, ... As. F § . The letters @,, dz,...@,1; can be chosen in (Ones) ways. When this set of letters has been selected, the number of forms corresponding to given values of 4, ps,... Mr depends (i) on what consecutive pairs of yw’s are equal, and (ii) on whether p, —2u,=0, 1 or >1. This number is then quite independent of 8, provided that 6S7r+1. Also the set of values which can be given to (4), #s,...4,) is independent of 6. Hence if ¢(7, w) is the number of independent forms P of degree & for which Zp=a, p,> 0, and Bra = Prag = +. = Pea = 0, then $(7, a) is the number of independent forms of degree 7+1, for which no yu is zero, and Suw=a. Now if y, is not zero, the number of forms of degree +1, corresponding to a given set of values (4m, fo,--.,) of the y's, is the same as the number of forms corresponding to the set of values (u,—2, so.—1,...4,—1). Hence ¢(7, a) is the total number of forms of degree r+1, for which S»=a—r—-1. Again, the total number of forms of degree 6 is equal to the number of forms for which p, is the last non-zero «, together with the number of forms for which ps, is the last non-zero w, and so on. Hence this number ; =$(1, =)(3)+4@ #)(8)+..+66-2 #)(,°))+66-1 =)(3). Now we shall assume that the total number of forms of degree <6 and weight w is ake 2741+r— *) e (ay r— az r—2 a 4 © « r—2 where w=: also that the number of forms of degree 8, for which Su=a', a’ . C is reducible if X=1, 2, 3, and n.=m=n; = %. Wot exe PART: 10 IV. On certain Quintic Surfaces which admit of Integrals of the First Kind of Total Differentials. Second Paper. By Anrtuur Berry, M.A., King’s College, Cambridge. [Received 31 October 1904. | CONTENTS. PAGE § 1. Introduction : ; : ; : : 5 é : : : - : 5 : 74 § 2. Quintics with a Aonble conic and a ‘ciple point. : : ; = : 2 : 2 : 75 § 3. Quintics with a double conic and a double point . : : : a 3 : . 76 § 4. Quintics with a non-degenerate double conic, but with no distinet shraltiple point. : 83 § 5. Quintics with a double conic, consisting of two distinct intersecting straight lines, but ath no distinct multiple point . Be ett : - 90 § 6. Quintics with a double conic, consisting of two coitieident staph Hines, iat with no distinct multiple point : ¢ - : ; : - : 5 : - 3 - : - - 104 § 7. Conclusion . ; : : . : 5 ; - : cs : : - : . ~ are § 1. Inrropucrion. THIS paper is a continuation of one with the same title presented to the Society on January 4, 1902 and published in Volume x1x of the Cambridge Philosophical Transactions ; it is closely connected with an earlier paper “On Quartic Surfaces which admit of Integrals of the first kind of Total Differentials,” published in Volume xvii of the same J'ransactions. I shall refer to these papers as “Quintics I” and “ Quarties.” The problem considered is the discovery of quintic surfaces which admit of integrals of the first kind (everywhere finite) of total differentials. In “Quintics I” I dealt with cases in which the surface possesses a singular line of multiplicity greater than two, or a double curve of order greater than two, or two non-intersecting double straight lines. The present paper deals with cases where there is a double conic, which may in certain cases degenerate into two intersecting straight lines, distinct or coincident. The cases not dealt with generally in either paper are accordingly those in which the surface possesses a double straight line and no other Lil Mr BERRY, ON CERTAIN QUINTIC SURFACES. 7 singular curve, or merely multiple points but no singular curve ; certain surfaces belonging to these latter types were however dealt with in a somewhat different connection in Parts II and III of “Quinties I.” In the present paper cases in which a surface has any further singular line in addition to the double conic, or in which the conic breaks up into two straight lines, one of which is a triple line, are ignored, as having been already dealt with. It is also unnecessary to consider cones. Surfaces linearly transformable into one another are not regarded as distinct, and, as a rule, specializations obtained by giving particular values to coefficients are ignored. Subject to these restrictions the paper professes to solve its problem completely, but owing to the complexity of the algebra I have very possibly missed some cases, though, I hope, not many. I consider first the cases in which the quintic has in addition to the double conic a distinct triple point (§ 2) or double point (§ 3), in which cases the surface can be birationally transformed into one of lower order. I then consider the cases where there is no such point, distinguishing for convenience the cases when the double conic is non-degenerate (§ 4), or degenerates into two distinct straight lines (§ 5), or into two coincident straight lines (§ 6). The equations of the surfaces discovered are marked with Roman numerals in continuation of the enumeration given in “Quintics I.” A summary of the results is given in § 7. § 2. QUINTICS WITH A DOUBLE CONIC AND A TRIPLE POINT. Since a quintic with a quintuple point is a cone and a quintic with a quadruple point is rational and therefore cannot admit of an integral of the first kind, we begin with the case of a triple poit not lying on the double conic. The surface can be written SF =PF2t+ quw + uw =0, where q, %, u; are homogeneous functions of a, y, z, of degrees 2, 2, 3 respectively ; and the double conic is g=w=0. Transforming birationally, by taking 2, y, z, g/w as new variables, we obtain the cubic Fz wet+ win t+u,=0. As a non-conical cubic is rational, this must reduce to a cone, and therefore F must be annihilated by a differential operator of the form d d d d dae Cagine +d—. 2 dz dw Substitutmg and equating to zero the coefficient of w? we have c=0; and since a, 6 obviously cannot simultaneously vanish we can by linear transformation of x, y reduce the operator to = +d = . Thus F is a function of y, z, w — daz, and hence F = (w— da) z+(w— dz) % 4+ 2s, where 22, v; are independent of 2, w. Retransforming, we find for the quintic S=(q — daw) z+ (q — daw) vw + vgw? =O oe.ceecseeceeeeeee (XXIII). 12 76 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF Let q = ax + (y, 20.2, Y, 2). Then there is a triple point where y= z= 0, az —dw=0; this is a second distinct triple point, if neither a nor d vanishes. If d=0, a+0, the two triple points coalesce into a triple point of a more complicated character, the tangent cone at which breaks up into the three planes v, = 0; the surface assumes the simpler form (Za GORING OI 30)" hogoncacnoosacseccoortonosednoc ut (XXIV). If a=0, d+0, the second triple point becomes the point y=z=w=0 which hes on the double conic; we have there a triple point of a complicated character, the tangent cone at which consists of 3 planes two of which coincide and moreover cut the surface in 5 straight lines. If both a and d vanish, y = z=0 is a triple straight line, so that we revert to a case already considered in “ Quinties I.” The four fundamental quadries 6 (ef. “ Quintics I,” § 1) are easily found to be respectively aq, —-$q—dzew, yqa—dyw, 2,—dzw, — swat $du’, dq where q, stands for ~ : The general surface (XXIII) which we have obtained is the general quintic with a double conic and two triple points, a surface considered by Cayley*, and shewn by him to be rationally transformable into a cubic cone. The cases when a or d vanishes are limiting cases. § 3. QUINTICS WITH A DOUBLE CONIC AND A DOUBLE POINT. Taking the conic to be q=(a, y, 2)=0,w=0 and the double point to be s=y=z=0, the surface is S= Fiat quow + usw? + vw* = 0, where 2, U2, Us, Uz, are homogeneous functions of «, y, z of degree indicated by the suffix. Neither w, nor v, can vanish identically, We assume that there is no triple point off w=0, as this would lead back to the case of § 2. Moreover, if there is more than one double point we assume that the most complicated one has been chosen to be «= y= z=0, where we count a tacnode as more complicated than an ordinary uniplanar point, a uniplanar point than a biplanar point, a biplanar point than an ordinary conical point. Thus we may avoid duplication of cases by rejecting all cases in which the surface has elsewhere a double point of a more complicated character than the one at w=y=z=0. By the same quadric transformation as in § 2, we obtain the quartic F = qv. + usw + usw? + uw = 0. We observe that from the nature of the quadric transformation F can only reduce to a cubic if v.=0, or if the conic split up into two straight lines, one of which is a triple line; both * “Qn the Deficiency of certain Surfaces,” Math. Annalen, Vol. un. (1871); Coll, Math. Papers, Vol. vu. p. 397. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 77 these cases can be ignored. Moreover, we need not consider the case when F’ becomes a cone ; for since the section by w=0 is two conics, either # degenerates, or these conics become pairs of generators; in this latter case by a linear transformation of x, y, z, F is reducible to a function of x, y, w and on retransformation we see that the quintic f=0 is also a cone. We have therefore only to consider the possibility that / may coincide with one of the known non-conical quartic surfaces which admit of integrals of the first kind*. If F be such a quartic it must satisfy the differential equation dF 6, dF eee ae @, aF _ dz dw . da nue dy 0, where 06;=a;4 + b:y+c;2+d;w, and a,+b,+¢e,+d,=0. The functions @ have also to satisfy certain conditions at the singularities of /'; in particular the conic 0¢,=0, w=0 must be an adjoint on the section of # by w=0; but this section is two conics, hence 6, divides by w, Le. a,=b,=c,=0. Any linear transformation of F which involves 2, y, z only corresponds to an exactly similar transformation of the quintic f and can therefore be employed freely, but a linear transformation involving w corresponds to a rational quadric transformation of f. Transforming first x, y, z we know from the theory of bilinear forms that the quantities 0 can be reduced to one of the three types Ge EC USER. (7 SC LOTR 9h aaonconeceseeccactaesoeoscaneopengoy (1); THe oO MTR. inte Gna arn (Peau UU) GopeoscooseesoadoodDocbede (11) ; ac+tdw, e+aytdsw, yrazt dw, Aw eie.cecrcrcccccccsenses (ii1) ; where aj, bs, c3, dy; Gh, Gh, C3, dy; Gh, G, &, dy; are now the roots of the fundamental determinant on which the integration of the differential equation depends. By the known results for quartics these roots taken in some order must be one of the four sets of quantities, 1,—1, 0,0; 1,1, -—1, —1; 3, -—3,:1,—1; 0,0,0,0. In type (1) the roots cannot all vanish or we should have a cone; in types (il), (iii) they must all vanish. Remembering that w is essentially different from 2, y, z, we thus have for a, bs, c;, d, one of the six sets of values 1; -1, 0, 0; 0, 0, 1, -1; if, 1, -l1, -1l 3h —3, 1 as ite il. -1, 3, —3; 0, 0, 0, 0. We have now to reduce further by linear transformations which involve w; either quoting the known results for quartics, or directly, rejecting cases in which the integrals of the funda- * “ Quartics,” p. 343, “ Quintics I,” p. 250 footnote, and De Franchis’s paper ‘‘ Le superficie irrazionali di 4° ordine di genere geometrico-superficiale nullo”: Rend. del Circolo Matematico di Palermo, t. xtv. (1900), pp. 1—13. 78 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF mental differential equation are logarithmic or conical, we get for the differential operator which annihilates # one of the nine forms: Ei 15 + 0-74 0-ae es ee (A 1), pet ee as) g etn : i , = Aig AOE 2 6 (B), SE ae Bn bo wo Wats danieeoanseacaesisoeen pec meeee (E 1), Ee 4p tS ge 31S iwde ee (E 2), Do +2 ite oo = RARBRE Anac Rcacocen capo rogsch tooo: (C), wa beg tO ge tO gd, diveldadsiscdveaatwaeedveeeeeees (D1), Ot tng t Og, do i. a (D 2), we thay t Vagt ay oWaligudeh sinsisee selec aceasta (F), where & », € stand for linear functions of the form #+Aw, y+ ww, z+ vw, and the reference letter corresponds to the notation which I have used for the several types of quartics which admit of integrals of the first kind (“ Quartics,” p. 343). Tt is known that the most general quartics satisfymg these equations actually possess integrals of the first kind, but may of course fail to do so if they are specialized. In our case the quartic is specialized by the conditions that the section by w=0 breaks up into two distinct conics, and that the point «= y=z=0 is an ordinary point on the surface. We have now to discuss these nine cases : (A 1). The fundamental integrals of the differential equation are &y, z, w and the most general quartic integral is: P= ly? + Ey (2, wl? + (z, wy The section by w=0 breaks up into conics of the form wy + h,2*, vy +h.2*; the equation may be written (En + ky2*) (En + kyz*) + En (2, w)' + (2, w) w = where the coefficients of the second and third terms have to be chosen so that the coefficient of w* may vanish. Taking g=ay+h,z*, ».=a2y +h, the quintic is F=U (ew +Aq) (yw + wg) + yew} {(aw + Ag) (yw + eq) + koz?w"}/q + (aw + Xq) (yw + wg) (eu, gy + (eu, @] = w=0. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 79 This has triple points where z=z=w+ Ay=0, y=z=w+prx=0; we thus revert to a case treated in § 2, unless \ = 7» =0; in this case the quintic is S= (ay + hez*) wt + xy (zw, 9)! w+ (2u, 9)? Z =O .....ce ccc ceeeeeee (XXV), the surface has triple points on the double conic, where x=z=w=0, and e=y=w=0, and may be regarded as a limiting case of Cayley’s surface (§ 2), when the two triple points coalesce with the conic. The fundamental quadrics 6 (“Quinties I,” § 1), are easily found to be aw, — yw, 0, 0. The surface thus belongs to the class considered in “ Quintics I,” Part II, and is in fact a special case of the surface (XVII). In general the conic is non-degenerate and the double point is an ordinary conical point. If k, vanishes the conic degenerates and if k,= 0 the double point becomes biplanar. If k,=k,=0 so. that in the quartic qv.=.*y*, we have the further possibility of taking q=H, v=y’, giving rise to an essentially distinct quintic S= {(wt rz) (yw + pe?! + @ (w+ Xx) (yw + pa?) (zw, 2) + (zw, x)}} +w=0 ...(XXVI), where as before the coefficients are so chosen that w divides out in the numerator. The double point is uniplanar, the conic degenerates into the two coincident straight lines x*=(, and there is a triple point on it where e=z=0. In general there is no other multiple point. The quadrics 6 are: —A#-—aw, 4rayt+5ua2+9yw, 4¢(Av+w), — bw (Ar +w), (A 2). The fundamental integrals are &, , w, and the quartic is F=(E, 0) + (& 0) Sw + Fw? =0, where the coefficients have to be so chosen that the coefficient of w' vanishes; 4, %, ws are all 0? along «=y =0, and therefore the quintic has an additional double line z=y=0. This case has already been considered in “ Quintics I.” (B). Three fundamental integrals are &/n, z/w, Ez, and F=(E, nz, wP=0, where as before the coefficients must be chosen so as to make the coefficient of w* vanish. The section by w= 0 is of the form (2, y)z*; this can be resolved into quadratic factors in three ways, giving three quadric transformations and therefore three quintics. If we take z to be common to gq and 2,, since it is also common to w;, it is a factor of the quintic, which accordingly degenerates; if we take g=2, v.=(2, y), then since uw, contains z as a factor, inspection of the form of the quintic shews that it has a triple line z=w=0. We have now only to consider the case in which g =(2, y)? = aa* + 2hay + by?, »=2: it is convenient to make a linear transformation of #, y so as to remove one of the constants 2, «, which occur in the @’s, say the former, so that & now reduces to z. We can now write F in the form (a, h, b\a, y+ pw)? 2+ (a, y+ ww) zw +(x, y + pw) ew? = 0, where we have expressed the condition that the coefficient of w* vanishes. We observe that w 80 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF cannot vanish, or the quartic would have a double point at «=y=z=0, which we have seen to be impossible, since w, cannot vanish identically. The quintic is Sz, h, b¥au, yw + wqy ew/g + (aw, yw + pq)? z + (aw, yw + wg) aq = 0. This has a double line along «= 0, w + buy =0, which is distinct from the conic unless b= 0, If b=0, the double line « = w=0, which is now part of the conic, is cuspidal with the fixed tangent plane w+ 2uhx =0, and this expression is also a factor of the terms which are cubic in a, w; thus the double line is tacnodal, being the limit of two double lines which intersect. We revert to a case already considered in “ Quinties I,” § 6. (E 1). The general integral is Fz ak? + b&w? + cEnfSw + dEw* + enf?=0. The section by w=0 is aa*y?+eyz*=0, which can only break up into two conics if @ or e vanishes. The birational transformation employed in “Quartics” § 5, shews that in each of these cases F’ is rational. (K 2), The general integral is F= ak? + bE + c&nfw + dnyiw + ef*w? = 0, where as before the coefficients must be chosen so that the coefficient of w* vanishes. The section by w=0 is aa*y?+ ba'z=0; if b=0 the quartic is rational as before, hence b+0 and may conveniently be replaced by unity; @ may or may not vanish. We now have two possibilities, g =a”, v.= 22+ ay? or g= 22+ ay’, » = 2". In the former case the quintic becomes S=la(w + ray? (yw + pr?) + v (w + rav)8 (zw + va?) + ca (w+ Ax) (yw + px?) (zw + vx) +d (yw t+ pax)? + ea? (zw + va?)?| + w =0, where On? + Av + cCApY + du? + ev? = 0. There is a uniplanar point at X::y:—2*; this is an isolated point and distinct from the double pomt at e=y=z=0, provided that X%+0; the surface thus possesses a double point of a more complicated kind than the one chosen as the principal point of the quadric trans- formation ; hence as explained at the beginning of this paragraph the surface can be ignored. If X= 0 this uniplanar point coalesces with a point on the conic ; f assumes the simpler form a (yw + wa?) w + @ (ew + va) w+ ca (yw + px”) (zw + va") +d! ly? (yw + wr? — pia? (cw + vary} /w = 0.0.0.0 (XXVIII). The double conic consists of the two coincident straight lines z= 0 and has a triple point on it where y=0; the double point «= y=z=0 is in general a conical point, but becomes biplanar if a= 0. The four quadrics 6 are —aw, 22, 3va%*+zw, 0. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 81 In the alternative case, g=az+ ay*, = 2°, the quintic becomes S= [a (ew + rq (yw + wg)? + (ew + AQ)’ (ew + vq)}/q + ¢ (aw + AQ) (yw + wq) (zw + vq) +d(ywt+ pg) t+e(zwt+rvgP gq] + w=0 oe (XXVIII), the constants being connected by the same equation as before. The double conic is in general non-degenerate, but reduces to two straight lines if a = 0; the double point at «=y=z=0 isa uniplanar point, the tangent plane at which is « = 0, and therefore touches the conic; there is a second uniplanar point, with the same property, at Aimivi—Av— ap’. To the generators of the cubic cone, into which the surface can be rationally transformed, correspond a family of rational quartic curves on the quintic, all passing through the two uniplanar points. The four quadrics @ are —2x2u+ 5rq —9aw, —2yu+10uq—4yw, —2z2u+1l5vqg+2w, 3wu+ 6w*, where u = 3Bvex + 4apy + rz. (C). The fundamental integrals are z, w, xz— yw; the quartic is F = (#2 — ywy + (xz — yw) (2, w)? + a (2, wy = 0, where we have expressed the condition that the coefficient of w* vanishes. The section by w=U is w(zon)s If we take g=2°, vy, =(z, «)', since @ is also a factor of ws, the quintic has a triple line along x=w=0; if we take g=(z, x), v.=2*, then since u, is quadratic in a, z the quintic has a double line z=z=0; lastly if we take « to be a common factor of q and 2, the quintic divides by x and degenerates. Thus this case can be ignored altogether. (D 1).. The fundamental integrals are z, w, #*—2yw; the quartic is F= (a — 2yw)? + (@ — 2yw) (2, wk +2 (2, w= 0. The section by w=0 is of the form (z*, z*; and the coefficient of w (us) is at least quadratic in z, 2; hence q, 2, uv; are all at least quadratic in z, z and the quintic has a double line along “z=z=0. (D 2). Ifthe constant » does not vanish it is easy to see that ~ can be removed by a linear transformation of «, y,z; if X=0 it is not generally possible to remove ». Thus we have two slightly different cases to consider. () »A+0, z=0. The fundamental integrals are aw, w, y*—2&z; the quartic is F=(y — 2&2? + (x? — 2&z) (x, wP +2 (2, we =0. The section by w=0 breaks up into two conics of the form y —2az+kh:=0 (G=l, 2) by a linear transformation of z, z we can reduce one of these conics, which we take to be q=0, to the form y*—2xz=0; the effect of this transformation is that the third term of F is zw (a, w)?. WOT eon AR TIE 11 82 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF The quintic is now f= qu (w— 2rzP + (w — 2dz) (aw, g) + w (aw, GP = 0. This has a triple point at «= y=0, w=2Xz and therefore belongs to a type already considered in § 2. (II) »=0. The quartic is reducible as before to F= (9? — 2x2) + (4° — 2x2) (a, wy + w(a, we =0, where the coefficients have to be chosen so that the coefficients of w* vanish, and q = y? — 2az. The corresponding quintic is F= [q(w? + 2uyw + wg)? + (w? + Qwyw + yg) (aw, gq)? + (ww, g)+w=0 ...XXTX). The double point z=y=z=0 is an ordinary conical point; the double conic is non- degenerate, and has on it a triple point, where s=y=0. The four quadrics 6 are: 4ya?, 4uny + 5ow, 4uaz+5uq + 5yw, — 6Gurw. (F). Three fundamental integrals are w, U = a2? -2yw, V =x — 3x2yw + 52w*; from which can be deduced V2— U3 Cae w = 6a'z — 3a*y? — 18xyzw + 8y>w + 92?w. The most general form of the quartic, subject to the condition that the coefficient of w* vanishes, is F=a¢+aU?+bVw+cUw=0. The section by w=0 is 6a%z — 32°y? + az, so that we can take g= 2", v, = 6a2z — 3y? + aa’, or vice versd. In the former case the quintic is f= (Gazw? — 3yw? — 18ayzw + 8yw + 92%a*) w + ax (w— 2y)P w + bx? (w? — Byw + 38xz) + cat (w—Qy)=0 «ee. (XXX). The double point at «= y=z=0 is a conical point, the double conic degenerates into the two coincident straight lines 2? = 0, and there is a triple point on the conic at y= 0. The quadrics @ are: a, —4ay+5aw, — 4x2 + 5yw, brew. If we take = 6x2 — 37°+ aa, v, = 2°, the quintic becomes f= {6e°2w® — 3a2yw? — 18ayzqu + 8y'qu + 92q@w + a (aw — 2yq)"} w/q +b (ew — 3ayqu + 329°) + ¢ (aw — 2yq)q=O0 ...... (XXXII). The double point is uniplanar, the tangent plane at it is e=0, which touches the double conic; the double conic is non-degenerate. There do not appear to be any additional singular points or singular lines. The quadrics @ are d — 2xq,+ 5g, — 2yq, + daw, — 22q, + 5yw, 3wq, where q = oe = 62+ 2az. This completes the enumeration for the case of a double conic and a double point not lying on it. We have found seven surfaces, given by the equations (XXV)—(XXXI). INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 83 § 4, QUINTICS WITH A NON-DEGENERATE DOUBLE CONIC BUT WITH NO DISTINCT MULTIPLE POINT. We write the quintic in the form Fz Pt + Qo + usu? + vw + vywt + Kw? =0, where v,, «, cannot both vanish, as we should then have a double point at w=y=z=0. In accordance with the general theory we have to find four quadrics 6; such that / satisfies the fundamental differential equation df fl df df _ 6, a Fe +6, + 455 ae a, | ted (A), while the 6@’s are connected by the he: ey dé, _d@, _ dO, _ d@, __ eit Fi am iam y SChenrpaconbcareCoeeeBaace eee EeAeOr (B). The @s have also to satisfy certain conditions at the singularities. Since the curve 6,=0, w=0 must be adjoint on f=0, w=0 (“Quintics I,” p. 265), and this latter curve is a double conic, @, must vanish identically when w=0, i.e. must divide by w. Further each of the six cubics x0, —y@,, etc., pass through the double conic. We thus get 6,=cutaqgt+wO,, O=yut+8q+uwO., O=2ut+yq+vO;, O=wvt+ dd’, where wu, v are linear in 2, y, z and O;=a;7 + b;y+¢;2+d;w (t=1, 2, 3). Equating to zero the term independent of w in the fundamental differential equation, we have Sug?u, + 4q°qath + F (a oe + ety ge) =0, where 2g, is used to denote dq, dq et B dy aay Vdz" It follows that (5~+4g.)u, must divide by qg, but q is assumed non-degenerate, therefore 5u+4q.=0, and Uh diy du, Canal agi daa Applying now the subsidiary equation (B) we find that v=§q, and a Dg Cy ig Oe cree cereie sis ss oe tise a loss sla os aoe eee (2). The quadrics @ now reduce to —4rq.taqg+wO,, —4yqa+8q+wO., —42q.+7q+wO;, Swqat Fy Og? arate (3), subject to the condition (2). The differential equation is much simplified if we can remove the constants ions, Oboe, (Oley ANN @,, ©,, @;. This can in general be effected by substitutions of the form x+)w, y + ww, 2+ vw, for x, y, z. We suppose now that this reduction can be effected, postponing to the latter part of this article the discussion of the case when this reduction is impossible. 11—2 84 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF Case I: dy, do, d, all zero. We proceed first to reduce @,, ©,, ©,. For brevity let us denote by A the differential operator d d d lee eayee haee Equating to zero the coefficient of w' in the fundamental differential equation, we have 0, d d ah A(qgiu)+¢ (ai +8 an =) Up = 0; hence Aq must divide by q, say where k is a constant which may vanish. By the theory of bilinear forms we can reduce A to one of the three types: d d he a + bsy a, Fo Fo seeeeteetetneeeeeetseeeseeeees seteeneny (i), d d d a: Qe Fi ae (a + ay) dy + C32 7B eevee reece ccn cence sssccsesccssucs (11), d d d ae he 7, +(e + ay) a +(y +42) dg (111). Employing these forms in equation (4) and remembering that q is not to degenerate, we easily find that in Case (i). k=one of a, bs, cs, Say a, and then b,+¢;= 2h, so that 2a, — b, —¢,=0. Case (ii). gq always degenerates. Case (iii). &=a, and gq is of the form aa* + (y?— 2zx), which is reducible by a substitution, z+nre# for z, to y? — 2za. Also from the coefficient of w* in the fundamental differential equation «d,=0. If «+0, d,=0; if «=0 then from the coefficient of w*, since v, cannot now vanish identically, one of a, + 2d,, b. + 2d,, c,; + 2d, must vanish; thus in all cases we have the alternative conditions a—0) 07 a, +-20,= 0) or b,--\2d,—10N orn cs 20, — Om eee eae (5). We now have enough equations to determine the ratios of the quantities a, by, ¢;, d,; we can obviously multiply them by any convenient numerical factor ; solving we thus obtain four sets of values: belonging to type (1) 0, 1, = 41, (U Bnrcae sapucnbobcadsagodoscaSimicicasc: (A), l, 6 =4. 29 4 ee (B), (or 1, —4, —6, —3) 0, 0, 0, Ole decneeitasdecece rca se sn cen teeeeee (C), belonging to type (11) 0, 0, 0, Qo nce ceckccnen davse aces cssneeecemenee (D). In the first three cases ‘we can take q=«—2yz and in (D) q=y?— 222. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 85 Tn cases (A) or (B) if we employ the transformation 282 + (c, — 4d,) w = 282’, which only fails if 8=0, A is transformed into an expression of the form , d 1, @ Lee! (ae + by) 7, + b,'y dy? CPt a2) 77 where the new constants ay’, 6.’, c,;', d, have respectively one of the two sets of values oreo | ? 3, q, — §, or 1 — 3, i, = 3, and the 6@’s are otherwise unchanged. The same reasoning as before shews that these new constants must still satisfy the condition (5). On substitution of the numbers we see that this condition is not satisfied. Similarly, if y+0, we can employ the substitution 2yy +(b.—4d,) w= 2yy’, leading to the sets of values which are similarly impossible. Hence cases (A) and (B) are impossible unless 8=y=0. In case (A) when B=y=0 we obtain from the coefficients of w* and w* in the fundamental differential equation ee d aan Gar.e+(y G2 q2) =O Aanv Pep ie yi - 25) m=0 ear kee (v3, de) a From the first equation we see that v, is independent of «, y, z, and from the coefficient of im the second equation we then see that a=0. The whole differential equation now reduces to d d (Yaya) /~° the integral of which is $ (a, w, yz) or, more conveniently (a, w, g), so that the quintic is Sz=FL2+9d@G “) w+a(q, 7) w+ (q, 2) w+ rA¥wwt + cw? =0......... (XXXII), where q=@ —2yz. The birational transformation q=YW, «=X, z=Z, w=W converts this into the cubie cone YX+Y(YW, X2)+X(YW, X*)4+(VYW, X*) W4+ AX W?4+ «W=0, which is in general non-singular, so that an integral of the first kind effectively exists. The surface has triple points at e=y=w=0,7=z=w=0. The surface satisfies the same differential equation as the surfaces (XVII) (“ Quintics I,” p. 277), and (XXV) of this paper (§ 3). It is a special case of the former and a slight generalisation of the latter: if we put X=«=0 we get (XXV). In case (B), when B=y=0, the equation to be integrated is (—Fau* + aq + ew) _ (— Far +6w) y 7+ (—$ eur — 4) 204 (Fax—5w) wT =0, where as before (=x — 2yz. 86 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF This admits the three independent integrals* ysut, U=qz, V=(2aq —daw) yzw, from which can be constructed h = (V2 — 50y%2*wt)?/ Uysztwt = y (4aq? — 20axrw + 25w*)?. The most general form of f is now SENGPZ + BOF VV HO oo. ccccecceseesneenenvoensenes (XX XIII). If we put 2aq —5x2w = Xu, 4aq® — 20aaw + 25w = Ww, q= Yu, z=Z, which is a birational transformation, we obtain the quartic 5OAY2Z? + (uW? + vXZ)(X?— YW)=0 This is a slightly specialised form of the surface # (“ Quartics,” p. 343), and as shewn there can be transformed into a non-singular cubic cone, so that an integral of the first kind effectively exists. The quintic has tacnodes (close-points) on the conic at 2=y—0; 2—2—0) In case (C), A=0. The coefticients of w* and w’ in the fundamental differential equation give the equations d d d gk=0, 49.,+9 \ an B dy +e 1) %,= 0. Since g, cannot vanish identically when qg is not degenerate, nor be a factor of g, the first equation shews that « =0 and the second that v,=0; thus we have an impossible case. In-case (D), d d AS ee g=Y¥Y — 222. From the coefficient of w® in the fundamental differential equation, we have dv, dv, 6 gare + 27 tae = ((). From the coefficient of z in this equation, we have «a=0, so that « or a vanishes. If « = 0, v, is merely a, and the coefficient of w* in the fundamental differential equation then gives av, dvs Ada . +qat(a Gy? yt) = 0. From the coefficients of y°, za a+2f=0, —G6a+2f=0, where 2f is the coefficient of yz in v, whence a=/f=0. Thus a always vanishes. The @s now reduce to — 42 (By—yx), —4y(By—yx)+Bqteu, —$2(By—ye)tagtyw, gw (By — 72). * In this and similar cases these integrals were by no means obvious, but, as when once discovered their verification is simple, I have not thought it worth while to give any indication of the process of discovery. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 87 If 8+0 the transformation 282 —w=2z' does not affect 6,, 6, but removes the terms aww, yw from @, @;; we thus revert to case (C) which has already been shewn to fail. Thus we must have B=0. The fundamental differential equation now has the integrals Bw, «aq, (2yy+w)/a, and the quintic is S=E (&, 2yy tw) +4 (a, 2yy+wPwt (a, 2yy4+wyYw=0 ...... (XXXIV). The birational transformation g= ZW, «=X, y= Y, w= W converts this into 2 (X, 2y¥+ WY 4+ ZX, 2V + WP+(X, 2Y¥+ Wy=0, a cubic cone which is in general non-singular. The quintic has a triple point of a complicated kind on the conic at «=y=0. Case II: d,, ds, d, not all zero. We have now to investigate the conditions that it should be impossible to remove d,, dz, ds, by linear transformation. If we substitute e , / 7) ‘ L=2 +=wW y=y+-—W, 2=2+ we Cae ae g W, @ where &, 7, € w are constants subject to the restriction +0, and 2’, y’, 2 are new variables, and express the fundamental differential equation in terms of «’y'z’w, the coefficient of w* in the new @,', ic. the new. d,, becomes {w6, (&, , 6)— &0,(& 7, €)}/m?, with similar expressions for d,, d;. Thus d,, d,, d; can be made to vanish if the three quadrics S,= 0,-—20,/u=0, S,=@,—y0,/w=0, S, = 0; — 26,/w =0, have a common point not lying on w=0. The process fails only if S,=0, S,=0, S;=0 meet only on w= 0, or in other words, if these equations, regarded as quadratic equations in the three non-homogeneous variables w/w, y/w, z/w, are inconsistent. We take the @’s in the form given by (3), (p. 83). It is convenient to distinguish three cases according as (I) the point 4, 8, y does not lie on the conic g=0, (II) 2, B, y lies on g=0, (III) «,8,y all vanish. (1) By linear transformation of #, y, z we can arrange that a=1, 8=y=0,and q=2*— 2yz. The equation (4) then gives wv (aye + by +42) — y (agu + dy + 052) — 2 (age + boy + 0,2) = 2h (a? — 2yz), whence SSO! CoS, Cy Gy Bhim pS Oy sce concouscanccctiosuacod (6). By a substitution (y+ uw, y) we can now remove ¢,, and by a substitution (2+ vw, z) we can remove b,; these substitutions having been performed, a substitution (w+ Xw, x) removes d,. The @s now have the forms: —4e+q+aqaw, —4ayt+b.ywt+dw, —4a2+0¢,2w+d,w, gaw+4hd,w 88 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF Accordingly 1=—-@-—2yzt+a/ru, S,=—2ey+b/ywt+dw, S,=—2a24+ c/ew + dw", where «a,’, by’, c;) are written for brevity in place of a,—4d,, b, — 44, c; — $d. The three quadrics S meet on w=0 in the two points z=y=0, =z=0, and nowhere else on w=0; hence, if the quadrics S are to have no common points off w=0, these two points must be multiple points of intersection together equivalent to 8 points. At «=z=w=0 the tangent planes to S,, S, are z=0, 2y—b.w=0. Hence if this point is to count triply as an intersection of S,, S., S;, the line of intersection must be a generator of the cone S;, therefore d;=0. Similarly if «=y=w=0 counts triply as a point of inter- section, d,=0. But by hypothesis d., d; are not both zero. We must therefore have one of d,, d; zero, say d,=0, and the point c=z=w=0 must count as 6 points of intersection of the three surfaces. Now «=z=0 is a line on S,, S; meeting S, where b.'y+d.w=0; this gives a finite value of y, contrary to hypothesis, unless b.)=0. Again x—a,w=z=0 is a line on S,, 8; meeting S, where — 2a,'y+dsw=0. Therefore also a,/=0, and then from (6) ¢,’=0, so that a,=b.=c,=d,=0. The 6’s now reduce to —4a°+q, —tay+dw*, —4xz, Saw, where d,+0, and may conveniently be taken to be 1. The fundamental differential equation now has the three integrals Sw, U=z(q¢+4ueu"), V=qtyt (Gaye — 42°) w*— Sewt. Moreover these can be shewn to be the only quintic integrals, so that the surface is FEB Ue Vi HO Siorceeeceeeee eee eee (XXXV). The surface has a tacnode on the double conic at e=2=0, the tangent plane there being z= 0. The transformation (q° + Gazqu? + 62w)/w= AW, (g?+4e2ew)/w=XZ, g/w=Y, 2=Z, which is birational, transforms 7 into AZ + uXZ+ sv (X*— ZW?) =0, a non-singular cubic cone. (II) Let a, 8, y lie on g=0, then we can arrange axes of 2, y, 2 so that g=a*—2yz, a=y=0, 8=1. The condition that Ag divides by qg shews as before that A can be reduced to the form d Mier d (qa st by + C2) aE + (cx + bY) ay + (b,x + Cs2) a ) where 20 = De = 65 SO} cezeenn cece octave sess sus sss sonseeeeee eee (7). Further by a substitution (#+)w, #) we can remove d,, and by a substitution (2+ vw, 2) we can remove d;. The @’s are now faz+(qe+by+ez)wtdw*, tyzt+qt(aqrt+by)wu, 424+ (be+ce2)u, —Sewt+hdw INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 89 The quadrics S accordingly become S, = 2a2+ (ae + byt+az)wtdw=0, S,=a+(q¢+b/y)w=0, S,=222+(be+e,/2z)w=0, where as before ay’, by’, cs’ stand for a,—4d,, 6,—4d,, c; -—4d,. If b,’+0, from S, and S, b, (2x2 + aaw + e2w + dw") — b, (22 + ¢,2w) = 0. If also b, +0 we can eliminate # from S,, obtaining a quartic in z, w in which the coefficient of 2* does not vanish; we find accordingly at least one finite value of z:w; then S, gives a finite value of ~:w and S, a finite value of y:w, so that S,, S,, S, meet at least once off w =0, contrary to hypothesis. Hence if b..+0,b,=0; S,; then factorizes into z.(2z2+¢,w). If z=0 then from S, a,x +d,w =0,so that we have a finite value of «:w unless a,’=0, and hence from 8S, a finite value of y:w. The second factor of S, gives z=—4c,’w and S, then gives 1 — xe, + w (d, —4¢,'), so that we get again a finite value unless c,,=0. If a,’=c,’=0 then from (7) b,’=0 also. Thus we must always have b,/=0. S,=0 and S,=0 are now satisfied by =z=0, and S, gives then a finite value of y:w unless b,=0. The quadrics S are now merely functions of «, z, w and have in general no common solutions. From b,’=0 and (7) we find a:b,:¢e;:d,=2:—3:—7:—6; and we may assume (a) that these constants have these actual values, or (8) that they all vanish. In case (a) the @’s are fxz2+(2r+q2)w+dw*, 4yz+q+(9e—3y)w, 424+ 72w, —S2w—-3w* It is convenient to make a substitution (7 +w, x) so as to remove d, thereby in general reintroducing d,; and then to make a substitution (y + Xz, y), (y + Ay + $2*z, y) which can be so chosen as to remove ¢, without altering g. The @’s are now 4az+2aw, 4yz+q—3ywt+dw, 42+472w, —S2w—3w’ The terms independent of # in the quintic f must now satisfy the differential equation 4 F d » a (— Syz — 38yw + dw’) = + (422+ Tzw) = — (£zw+3w”) _ =0. Two independent integrals of this equation are 3(z2+ 5w) wt, 10y/w + d, log (z+ 5w)/z. It is evident that from these integrals no quintic integral can be constructed, d, being by hypothesis not zero, so that the quintic surface divides by z and therefore degenerates. In case (8) the 6@’s are faz+qzwt+dw, 4yz+qtorw, #2, —S2w, when d,+0 by hypothesis. Mor SOS Par I 12 90 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF We now remove ¢, by a substitution (x, w), thereby reintroducing d,; we can then remove d, by the substitution (# +z, x), (y + Aw +42*z, y) used in case (2). Thus we have removed c, from 6,, 6, without otherwise altering the form of the @’s. The fundamental differential equation now has the three independent integrals: Sur, U=(4ez+dyw*) zw, V = 6q2w + 6d,zw* + dw". From these we can construct also the quintic integral 6 = {V?—d,U3/2w*}/Aw? = 4 {9q?2 + 18d,gaw? — 16d,2°w* — 6d,eyw'}. It can be verified that these are the only quintic integrals, so that our surface is f=rBu? + wU +0V + pWH=O........ceceeseseec eee eeees (XXXVI). The birational transformation EV Vi Wee — ee — converts the surface into AZ + pYZ*? + vXZ?+ p (X*Z—d,Y*) =0, a cubic cone which is in general non-singular. The quintic has a triple point of a complicated type on the conic at e=z=w=0. This completes the case when the double conic does not degenerate and the surface has no singularity distinct from the conic. We have found five surfaces represented by (XXXII)— (XXXVI). § 5. QUINTICS WITH A DOUBLE CONIC, CONSISTING OF TWO DISTINCT INTERSECTING STRAIGHT LINES, BUT WITH NO DISTINCT MULTIPLE POINT. We take the double lines to be zy=0, w=0, so that the quintic is of the form f= yy + cyuw + uw + vs + vw + Kw? = 0. As at the beginning of § 4 we reduce the @’s to the form wut+ary+Ow+dw, yut+ Bayt+ Ow+dwu*, zu+yayt+ Ow+dw, we+tdw*, where ©;=a;2+b;y+¢;z, and u, v, are also linear functions of a, y, z. Now the conic 6,=z=0 must be adjomt along the section of f by «=0, and as this section consists in part of the double straight line z= w= 0, the conic 6; = «=0 must reduce to w=0; Le. b}=c,=0; similarly a,=c,=0. Similar reasoning shews that for all values of X:m the conic O,+40,=0, Awx+ ww=0 must be 0? along 2=w=0, *, e(—v+u)+ aay must be 0? along z=w=0, “. v=utayt+ a2, and similarly v=ut+Ba+d'y, J. v=ut Bet ay. The subsidiary differential equation (B), then gives 4u+82+ay+v=0, whence u=—-2(Betay), v=% ooo -_~ @ S f} 2 SS — INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 91 As before we have du, du, du, at eee and -0, Oy Cy + y= 0 .ccccscecccscssces (8). The @’s are now: —2x(Bu+ay)+axy + acw + dw", —2y (Ba + ay) + Bay + byw + dww*, —22(Ba + ay) + yryt (age + bay + e532) w + dyw?, Sw (Ba + ay) +4d,w’. As before we can remove d,, d., d; if the three quadrics S,= 6,-—20,/w=0, S,=0,—y0@,jw=0, S;= 0; —20,/w =0, have a common point off w=0. Now these quadrics are S,=—Pe+a/ew+ dw, S,=—ay?+ byw + dw, S, =—2 (Bx + ay) + yoy + (au + byy + ¢;'2) w + dyw? = 0...(9), where as before a’, b,, c; are written for a,—4d,, b,—4d,, c,—4d,. Hence we can remove d, in two ways if 8+0, and d, in two ways if a+0. It is convenient at this stage to distinguish three cases according as (I) neither a nor 8 vanishes, (II) one only vanishes, say a, (III) both vanish; and to subdivide the third case again according as (III A) y+0, or (III B) y=0. Case I: a+0, B+0. We can remove d,, d,. We suppose this done; then «=y=0 lies on S,, S, and meets S, where c,;'z + dw =0; thus we can remove d; unless c; =0. If ¢, = 0, then #=0, — ay+b,w=0, lies on S,, S, and meets S; where —b,/z + (b,b.'/a+d;)w=0; thus we can remove d; unless also b,’= 0, and similarly unless a,,=0, Thus we can remove d; unless a,’ = b,’ = c; = 0, whence ae: Take first the case, which proves to be simpler, in which d,+0, and may conveniently be taken to be unity, and a,=b,=c,=d,=0; we can further remove y by a transformation (z+ ra, x). The @’s are now: —2a(Bxrt+ay)+ary, —2y(Ba+ay)+Bey, —22(Brt+ay)+(act+byw+w, tw(Ba+ay). The coefficients of w*, w*, wi, w* in the fundamental equation now give the equations: d qa 3K (Bx + ay) =F (asx ae by) = V+ = V2 = 0, d d d d 2 (Bx + ay)v, + xy (« rss B5,) V, + (asx + byy) qe tt ae 0, d d d d (Ba + ay) v2 + vy (« FEE + B Al V2 + (30 + by) F, Us + Pe 0. From the first we infer that v, is independent of z, from the second and third that v, and w; are at most linear in 2. 12—2 92 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF In the last equation the coefficients of 2*z, y°z, xyz give the equations g=9, f=9, 2ag + 2Bf + c=0, where 29, of are the coefficients of wz, yz in v, and ¢ is the coefficient of 2 in wp. Hence c=0, so that the whole quintic is linear at most in z, and is accordingly rational or a cone. We thus revert to the second case, in which d, can be removed. By a substitution (¢ +z, x) we can remove a; unless ¢;= a, and similarly we can remove b, unless ¢, = by. Let us suppose jist that we have removed ag, J. If we make the transformation B«—(a,—4d,)w= 8x, we do not reintroduce any of d,, d,, d;; we may introduce a term b, yw in 6;, and the constants a, b., c;, d, acquire new values, say a secondary system, viz. these are a, —4(a,—4d,), b,43(aq—4d,), e,—2(a—4d,), d,+8(a,—4$d,). Similarly by the transformation ay—(b, —4d,)w=ay' we replace a, b,, ¢s, dy, by a tertiary system +3 (b.—$d,), b,- $(b,—$d,), c;—3(b,.—$d,), dyt+$(u—3d,). Now by considering the coefficients of w®, w* in the fundamental differential equation, we obtain as before (equation (5) of § 4) the alternative conditions a, + 2d,=0, or b;4-2d,= 0); or ic, -2d,—O0) Ord, —Oe--es-cen ee enaees (10). Since this condition is independent of the existence of a, and 6, it must be equally true for the secondary and tertiary systems of constants. For the primary system we need clearly only consider the three alternatives a, + 2d,=0, c,+ 2d,=0, d,=0; but for the secondary and tertiary systems we must consider all four. Applying the conditions (10) to the primary and secondary systems we obtain equations sufficient to determine the constants; on solving them we get nine possible systems, viz. : 0, 1 al 0 (a) ee oe 2. 0 (11) ib. 1, -2, 0 (iii) Ag pela 9, —2 (iv) 2, ieee ae re | (iW) pee oes steven ete sn eee (I) ep An 6, —3 (v1) 37° = 8, —4 (vil) il 1, -4, 2 (vill) 0, 0, 0, 0 (ix) In the last two cases the secondary and tertiary systems are the same as the primary INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 93 system. In the remaining seven cases the values of the tertiary constants (omitting common numerical factors) are respectively 3 here 6 () 2-3 —-— 8, 9 (ii) SS |G 3 Gi) 2, -—3, —13, 14 yy) SS Siena scat (II). Ga o= oe) == 8 4 (v) lp 1, -14, 12 (vi) 0, 0, i 1 (vil) We now see that the conditions (10) are only satisfied in cases (iii) and (v). In both these cases the tertiary constants are such that the constant a, if it has been introduced can be removed by a substitution (2+ Aw, z). Thus the tertiary system can be treated as the original primary system. Repeating the first transformation in these two cases we get the two sets of values 1, 1, —14, 12 and 0, 0, 1, — 1 which obviously fail. Thus all the cases (i)—(vii) fail, and we have only to consider cases (vili) and (ix). Now if the quintic is neither rational nor a cone it must contain a term at least quadratic in z. The terms highest in z in the quintic satisfy the fundamental differential equation with the constants y, a3, b; omitted, since the corresponding terms in the differential operator lower the degree in z; among the terms highest in 2, those highest in w satisfy the equation i, al d (tiga t Ont ge tate, WP Saree ncecese-oocbereecnoceness (11). The possible terms quadratic or cubic in z are from the form of f one or more of ay2w, «2°w’, yzw*, 2w, zw; and it is evident that in case (viii) none of these can satisfy (11). Thus we are reduced to case (ix). In this case the coefficient of w® in the fundamental differential equation gives x= 0, and then the coefficient of w* gives d d d 2 (Ba + ay) ntay(a oe +8 pe Y 5) v, = 0, which is impossible since the first term cannot divide by ay. Thus if d,, d., d, can be removed and also a,, b, can be removed there is no solution. Let us suppose neat that one of a3, b,; can be removed, say az, but not 6. Then c;=6,; we may still use the first transformation just employed, viz. {@c—(a—4d,)w, Bz}, so that the possible systems of values of the constants a, b,, c;, d, are the 9 just given in tabular form. But in none of these but the last is c;=0),; so that if a,=0, b,+0 all the four constants must vanish, Similarly if a,+0, b,=0. If neither a; nor b; can be removed then c,=a,=6,; and then the conditions (10) shew that we must also have a, = b,=c,=d,=0. We can now employ a transformation (¢+ Aw, z), to remove a; or b, without introducing d; and so reduce this case to the preceding one in which only one of a;, b; does not vanish. 94 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF In this case the terms in f which are independent of y satisfy the differential equation d d d (- Qu AE — 2z AE + 307) f= 0, of which the most general integral is («, 2)’ w*. Thus the term «w* vanishes, v;, v, divide by y and the former may be taken to be simply y. s The coefficient of w! in the differential equation now gives d d d d 2 (Bx + ay) + xy (2 Sas es tra) yt by G2 = 0. Everything in this divides by 7° except 38xy, therefore 8=0 contrary to hypothesis. The whole case when a+0, 8+0, accordingly leads to no solution. Case II: a=0, B+0. We can always remove d,; we can remove d, unless 6, —4d,=0. Let us suppose first that we can remove d, as well as d,. The quadrics S are now, d, and d, having been removed, S,=— Be+(a—4d,)ew, S,=(b.—4d,) yw, S, = — Buz+ yayt {asx + byt (es —4d,) 2} w + dyw*. As before these meet on a point off w=0, so that d, can be removed, unless a,=c;=}d,, whence a : bs: ¢;: dy=1:—4:1: 2. Let us first suppose that d, cannot be removed, so that the four constants a, Dace (a) are 1, —4, 1, 2, or (f) all vanish. In case (4) the 6’s are —2Be+aw, 2Bay—4yw, —2Byz+ yey + (asx + by +z)w+dw, 28aw+w* We can then remove b, by a substitution (z+ Ay, 2) and a; by a substitution (z+ Aw, 2). The terms in the quintic which are independent of y now satisfy the differential equation \- 2 Ba? + vw) i + (— 2 Baz + zw + dw?) = + (2 Bxew + w*) at =0. This has the two independent integrals 6 log #w*—5w/x, 282/«—d,(w/a), from which no quintic integral can be constructed, so that the whole quintic divides by y and therefore degenerates. In case (8) we can remove a; as before and we can remove y by a substitution (2+ ry, z) The 6’s are now —2B8a*, 2Bay, —2Baz+bywt+dw*, 2Brw. The fundamental differential equation now has the two obvious integrals w/w, ay’, and it can be verified that a third integral is {28az —(b,y+dyw) w} y. The only quintic integral which can be constructed from these is (x, w)?y¥’, so that f degenerates. We thus revert to the case in which d, can be removed, if d,, d, can; and consider it next. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 99 We can always remove a; by a substitution (2+Az, z) unless c,=a,. Also since d,, do, ds have been removed we have the conditions (10). Combining these with c,=«, we find that the constants a,, bo, c;, d, have one of the four sets of values Ohne OR Ou 0". Gs) 1, -—2, +1, 0 (x) Zee FoR Ne, ) Be di re) el Gis) If a; is removed we can use the first transformation employed in Case L, viz. {8x —(a,—4d,)w, Bz}, and by the same argument as before arrive at the values given in table (I). In Case I., however, we omitted the possibility b,+2d,=0, on account of symmetry; at present we no longer have symmetry and must therefore take this possibility into account. This leads to three new sets of values, viz.: Le Pee 2g, wg ‘gel 3; 8, —7, —4 AGXALV))i Pate ions aircrarartant cease oats (IV). Al * 4, 1, 2 (xv) Also as before one of the terms 2ryw, 2xw’, 2yw*, 2w*, 2w*? must exist in the quintic and must satisfy the equation md d Cs a a a (a FEE boy me + O52 7 + 3s a) f= 0; i.e. one of the five constants d+ b,+2¢,+4d,, %+2ce;+d,, b,+2e,+d,, 3¢;,+d,, 243d, must vanish, or more simply one of 2e;—ds, b:—¢;, h—Cs, 3¢+ds, 4e, + 3d,, must vanish. This new condition is satisfied in all cases of table (III); otherwise only in (ix) of table (I) and (xv) of table (IV). Thus the only possible sets of values of the four constants are those given in table (III), where in cases (ix) and (xii) a; may or may not vanish; in cases (x) and (xi) a, cannot vanish. In case (x) we can remove 6, by a substitution (z+ Ay, 2), and then the @s are —2PBe+aw, 2Bry—2yw, —2Raez+yeyt+aaew+zw, 2Baw. The fundamental differential equation then admits of the three independent integrals (Bz—wyw, xy(Bxr—w)w, 22/4 + yy/w+a, log y/w. The letter z only occurs in the last integral, which is logarithmic; hence the only algebraic integral must be independent of z, and the quintic is a cone. In case (xi) we have similarly two algebraic integrals which are independent of z and a logarithmic integral, viz.: ey, «(Be-—Zwyw, 52/2¢4+ yy/w + a, log y/w. 96 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF In case (xii) it is simpler to consider the differential equation satisfied by the terms in the quintic independent of y, viz.: d = Chere re \c 2 Ba? + aw) ag (— 2 Buz + a,rw + zw) apt (aw + w*) ait l= 0. This has the two independent integrals (Bz —a,w)/z, — 5w/a + B log aw. From these evidently no quintic term can be formed, whether a, vanishes or not, so that the original quintic divides by y. In case (ix) we can remove y by a substitution (¢+Ay, z) and a; by a substitution (z+rw, z); the 6’s then reduce to: —2Ba°, 2Bay, —2Baz+byw, 2Baw; and the fundamental differential equation has the three independent integrals wy, y/w, U=(2Ba2—byw)/2. If b,+0 the only quintics which can be constructed from these three integrals are x(y, wy, Ux(y, wy, so that the quintic degenerates. 3 If b,=0, the general quintic integral is (z, zQy, w), so that #=z=0 is a triple line, a case previously considered. This completes the case when a=0, 8+0, and d,, d, can be removed; and there is in this case no solution of our problem. We have now to consider secondly the case when a=0, B+0, d,=0 but d,+0; then as we have seen b, —4d,=0. We make no assumption for the present about d;, and express first the condition that a term quadratic or cubic in z exists. The highest term in z as we have seen must satisfy the fundamental differential equation modified by the omission of the coefficients y, as, bs, ds. If z°w? exists we have at once 3c,+ d,=0, whence a, :b.:¢;:d,=7:—3:2:—6. If z*w? does not exist there must be quadratic terms of the form cayZ?w + (ye + Koy) Zw? + C2 w*, Applying the differential equation and equating coefficients we obtain the four equations c (b, — es) — Bx. = 0, cd, — x, (b, — cs) + Be = 0, (3b; + 2cs) Ko = 0, dyke, + (3b, + 2c,) = 0. These are easily seen to be consistent only if b,=c;, or 3b,+2c;=0, whence 0, 3b,:03:0,=—4:1:1:2, or =38:—2:3:-—4. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS, 97 We thus have altogether 4 possible systems of values of the constants a,, b,, ¢, dy, viz.: 7, —8, 2, -—6 (4), — 4, 1. 1 2 (a1), 3, — 2, 3, -—4 (iii), 0, 0, 0, 0 (iv). In case (i), since none of c;— 6s, c;—4d;, c;— a, vanishes we can successively remove bs, ds, a3; the @’s then reduce to —28e°+Trw, *Pry —38yw+ dw, —2Baz+yeyt+2z2w, 28xw —3w’. 0 — 10w The fundamental differential equation has a logarithmic integral d, log —— = — 10y/w, and two algebraic integrals U= a5 (8x — 10w)' wt, V= (yay — zw — td.yaw) ew (Bx —10w). The quintic must be a function of U, V only; if y+0, V is the only possible quintic integral and the quintie accordingly degenerates. If y=0, V reduces to #zw*(8x—10w), and from this and V we deduce 2*w*, but no other quintic terms, so that again the quintic degenerates. Cases (ii) and (iii) can be interchanged by the transformation Bx—(a,—4}d,)w = Be’, already employed more than once; so that we need only consider one of them, say case (ii). We cannot in general remove d, or b,, but can remove a, by a substitution (z+, z). The terms independent of # in the quintic / satisfy the simplified differential equation yy wy) ‘ ae ytdw)o tet by + dyw) 7 +wa =0. This has the two independent integrals yjw—d.logw, bs (y/wy + 2dyy/w — 2d.z/w, of which the first is logarithmic and the second is of zero dimensions. Hence no quintic integral can be constructed from them, and the whole quintic degenerates by having « as a factor. It remains to consider case (iv) in which the constants a, b,, c;, d, all vanish. In this case we can remove d; by a substitution (z+ Ay, z), and then a, by a substitution (z+ pw, Zz); and the 6's are then: —2Ba?, 2Pay+dw, —2Baz+yay+ byw, 2Baw. The corresponding fundamental differential equation has the three independent integrals yu, U=(Bry—dw*)aw, V = {Ratz — Byaty —4Rb,cyw + ¢ydcw + Lb.dyw*} w. From these we can in general construct no quintic integral which is quadratic in z, so that the surface is rational or a cone. But, if y=b,=0, we have an integral z/x and the general form of the quintic is then f= (Bary — dw’)? (a, 2) + (Bay — dyw*) (a, 2)? w+ (a, zPw?=0...... (XXXVI). Vor XX. Parr I: 13 98 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF The transformation Bay —dw*= YW, «=X, z=Z, w= W converts this into ¥2(X, Z+Y(X, Z+ (xX, ZP=0, a cubie cone, which is in general non-singular. The quintic has a triple point at # =z=w=0; and is further characterized by the existence on it of a family of pairs of conics, which are cut out by planes of the family #+Az=0. These conics correspond to the generators of the cubic cone into which the surface has been transformed. We have now completed the discussion of the case a= 0, 8+0, and have found merely the surface (XX XVII) just considered. Case Tif a> a—B—0; y=-0: We may conveniently take y to be unity. : Ate : : : d From the terms independent of w in the differential equation we have “1 = 0, so that dz u, is independent of z. From the coefficient of w in the differential equation we have du d d Gi) Sees Es Ya + {ase Aah by a + (asx + bsy + 652) =} ph (0), so that w, is at most linear in z; and similarly from the coefficient of w* we infer that wu; is at most quadratic in 2. Now we have seen that at least one term in the quintic must be quadratic or cubic in z ; hence the only possible terms of this type are «,xz*w", «y2*w*, ¢2°w*. Hence as before we must have one of the three alternative conditions Oy 2e3;4d, =cs—b, = 0) c — a = 05) 44-Sd = 0) secaeeteeceeeneee (12). We now consider as before the possibility of removing the constants d,, d., d; by substitutions (7+ rw, x), (yt pu, y), (2+ vu, 2). First: let d,;, d. be removed. Then the four fundamental constants a@,, b., cs, d, cannot : f df ce vanish, since we should then have a 9, giving a cone. dz The coefficient of 2? in the fundamental differential equation gives (G—b;)\q=0, (G—a;)K—0, (4¢,--3d,)e—0) cicn..-sccnnsecmeeeee (13). Also the coefficient of wyzw* in the differential equation gives C + gk» + bax, = 0. The last equation shews that if «,, «, both vanish ¢ vanishes also: therefore one of ,, Ks must not vanish, say x, and therefore from (13) INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 99 We can remove d; by a substitution (z+ vw, z) unless c;=4d,. If this equation holds, then the four fundamental constants are — 4, 1, 1, 2; since c,+a, we can remove a, and the 6’s are then: —4aw, yw, «ryt+(byt+2)wt+dw, w* The fundamental equation now has the three independent integrals cut, yw, z/w+tay/w?— (by/w + d;) log w. The first two are independent of z; hence the third must occur in the quintic, and must therefore be non-logarithmic, ie. b,=d,;=0, and the third integral reduces to (52w + ay)/w®. Any quintic integral constructed from these three integrals, of which the last two are of zero dimensions, must have « as a factor, so that the quintic degenerates. We thus revert to the case c;+4d,, in which d, can always be removed. The coefficients of w° and w* in the fundamental differential equation then give as before the alternative conditions a, +2d,=0, or b,+2d,=0, or c,+2d,=0, or dy=O0.........cceeeees (15). The first, with (14), gives c;=4d, and can therefore be rejected; the second or third gives for the four fundamental constants the values (a) 3,— 2, —2,1 and the last gives (8) 2,—1,—1,0. In case (a) we can remove a; by a substitution (z+ Az, z), and the 6’s are B3cw, —2yw, «y+(bsy—2z)w, tw We now have the three integrals of the fundamental differential equation ay, yu, — zut+ 2ryw* + 2b,yw* log w. As before, since z must occur in the quintic, the last integral must be non-logarithmic, so that b,=0, and the quintic is then f=2 (Qay—5zwP + y (xy, w? + (Qey — 52w) (xy, w?) w=0...... (XXXVI a). The quintic is quadratic in z, w?; hence the double line x =w=( is tacnodal and can be regarded as the limit of two coincident double lines; we therefore revert to the case of three intersecting double lines, two of which coincide, a case already treated in “Quintics I,” § 6; the surface which we have found is in fact reducible by the linear substitution Nm] ta VT Ve to the surface (VII) of that article. . In case (8) we can remove a, as before, so that the @’s are 2ew, —yw, rcyt+(by—z)u, 9. The differential equation has the integrals x7*, w, -—1a/w+z/y+b,logy. As before 6; must vanish ; and we now obtain - f= ax (ay — 2wz)? + bry (ay — 2wz) w + cay?w? + Kw = 0, so that = w=0 is a triple line, a case already considered. This completes the case when a=8=0, y=1 and d,, d, can be removed. 13—2 100 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF Secondly, let one of d,, d, be removable and the other not: let us take d, = 0, d, +0, then b= +0). We can always remove a; by a substitution (y+)w, ¥). The conditions (12) now give three sets of values for a, bs, cs, d,, viz., (a) —4, 1, 1, 2, (8) 3, = 2, +3, — 4, (y) 0, 0, 0, 0. In case (a) we can remove d, by a substitution (2+Ay, 2); the terms independent of # in the differential equation then satisfy the equation d d lw a dv) 5, + (bsy + 2) a +w iatt=9 3 this equation has the two independent integrals y/w—d,log w, 2d.z/w — byy?/w*. From these no quintic integral can be constructed, so that the original quintie divides by « and degenerates. In case (8), since c,+ b, or $d, we can remove b,, d; by substitutions (2 + ry, 2), (2 + ww. 2), and the @’s then reduce to sew, —2yw+dw, «y+3zw, —2w*. The fundamental differential equation now has the integrals aw’, 2y/w+d,logw, aw (ay? — 2d,zw*). The only quintic integrals which can be formed from these are simply the first and third of these, so that the quintic degenerates. In case (vy) we can remove d; by a substitution (2+ dy, z), and b, by a substitution (a@+pw, x), so that the @’s are Os choi, mo, (O The integrals of the fundamental differential equation are now a, w, xy? —dzw*, from which no quintic integral which is quadratic in z can be constructed. Thirdly, let neither d, nor d, be removable ; then a, = b,=}$d,, whence by (12) m, bs, ¢s, ds all vanish. We can remove as, b, by substitutions (#+ Aw, «), (y+mw, y) and then d; by a substitution (z+va, z). The 6’s are now CHO CET geo, Oh The fundamental differential equation has the integrals de —dy, w, 6dyvzw*— 3d,ay + dix, from which no quintic integral quadratic in z can be constructed ; so there is no solution of our problem. This completes the case in which a=8=0, y+0. We have found no new solution of our problem. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 101 Case IIT B: a=B=y=0. The @’s all divide by w, and we may for convenience omit this factor. The terms independent of w in the differential equation give the equation d d d) 2 pee » 3) = {20 + 2b,+ ax ae + boy a + (su + byy + 632) da| u, = 0, whence we obtain the alternative conditions 3a, + 2b,= 0; or 2a, + 8b,=0) Or C4 20g =O. ose escsseceesses cess (16). We have as before the conditions necessary for the existence of a term, quadratic or cubic in 2, 2¢,— d,—=0) or ¢;=4,, or c,=6,, or 4e, + 3d,=0; or 34, +4,=0 ......05: (alia): Solving these two systems of equations we get ten possible systems of values of q, ba, C3y ds, VIZ. : eae ee ee (i), Gen iG) tye 2. 4, Gi); Dae ot Soe Gait): ae ed Dee a) bs ifiy): Te ee VE eS Ge 29 12 2B) ae aeiGui), hee 6h a Cri): Se Bie Api te, he Guat), Te 518 terse ON tk (ex); 0, 0, 0, 0 (x), where the systems with the even numbers (ii), (iv), (vi), (vill) only differ from the immediately preceding systems by an interchange of x and y. We can now consider the possibility of removing the constants d,, d., ds. (I) Let d,, d., d,; be removable; then from consideration of the coefficients of the two highest powers of w in the differential equation we have as before the alternative conditions a,+ 2d,=0, or b,+2d,=0, or c, +2d,=0, or d,=0. On inspection we see that these are only satisfied im cases (ili) (or iv), (vii) (or viii), (ix), (x). But the last leads to we 0, giving a cone. dz Case (iii). We can remove b, by a substitution (2+ Ay, z); the @’s are now 2x, —3y, ase+2z, —4w. 102 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF The fundamental differential equation has the three imdependent integrals vy, «cw, a,log «—2z/x. In order that z may occur the last integral must be non-logarithmic, so that a, =0, and the integral is z/a. The most general quintic is now a sum of terms of the form (a, z)* y’, (w, z? yw®, (#, 2) wt; expressing the condition for the existence of the double conic we have therefore for our quintic f=vy («, 2) +y (a, zPw?t (a, z)uwt=0. This surface, like (XXXVII 4), has a tacnodal line y=w=0. If the coefficient of ay? is not merely # it can be taken to be z, and we have the slightly simplified form f= yz + y (&, ZP Ww? + (a, ZY WE Oo. erereeerereneee (XXX VIII a). This is easily seen to be identical with surface (VII), of “Quintics I,” § 6. But if this coefficient happens to be # we have the special case fz=vy' + y (@, zw? +(@, 2) wt=0 which was omitted by error in “Quintics I,” § 6. POE rca (XXXVIID), The birational transformation «= X, z= Z, w= W, y= W?/Y, converts this last surface into A? + V(X, ZP+ V?(X, Z)=0, a cubic cone which is in general non-singular. Case (vii). We can remove a;, b, in the usual way, so that the 6's are 4aw, —6yw, —zw, fw We have the three independent integrals a*y*, yw, 2w*, and the quintic is f=xy? + (aay + bz?) zw? + cywt =O crerceccreccerseneees (XX XIX), The surface has a triple point at c=z=w=0. The birational transformation «= X, z=Z, w= W, y= YW/X converts it into X?Y?+ aX VYZW+bXZ+ cY W*=0. This is a special case of the quartic which I call (E) (“Quartics,” p. 343); and the transformation there employed converts it into a cubic cone which is in general non-singular Case (ix). We can remove a;, b, and the 6’s are merely aw, —yw, 0, 0. The general integral is $(xy, 2, w) so that f=vyre + ay (cy, 2) wt2 (ay, 2P wt Acwt+ KWH 0 w..cereoeee (XL). INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 103 This surface is a specialization of (XXXII), of § 4 [see p. 85]; the double conic which was there assumed non-degenerate being in this case a pair of straight lines. (II) Let d,, d, be removable, but d, not removable. Then c,=4d,, and we have one of cases (i) (or ii), (ix), (x): and the last fails as before. Case (i). We can remove a, 6; so that the 6’s are proportional to 62, —9y, zt+dyu, w. The differential equation now has two algebraic integrals involving 2, y, w only, and a logarithmic integral involving z, so that we have a failure as before. Case (ix). We can remove a;, ); so that the @’s are proportional to zc, —y, dw, 0. This fails for the same reason as case (1). (III) Let one of d,, d, be removable, the other not; say, d,=0, d,+0, then a,=4}d,, and we have to consider cases (v) and (x). Case (v). We can remove a; by a substitution (z+, 2), thereby changing d, and then we can remove d; by a substitution (¢ + vw, z). The 6’s are now proportional to 2e+dw, —3y, by—3z, 2w. The differential equation has the two algebraic integrals d,z/y—b,2/w, yw, and a logarithmic integral 2a/w—d,logw. The only quintic integrals which can be constructed from these are of the form yu, (dzw—b «yew, (dzw— bay) yw, all of which divide by w, whether 6, vanish or not. Case (x). We can remove d; by a substitution (z+ 2z, z). The 6’s are now proportional to dw, 0, asx+by, 0. The differential equation has the integrals y, w, U = 2d,zw — a3x7 — 2b,vy. The most general quintic which can be constructed from these is f = U? (Ay + ww) t+ U(y, we + (y, wy =0. But since y=w=0 is to be a double line either X=y=0, or a,;=0, and in either case y=w=0 is at least a triple line. (IV) Let neither of d,, d, be removable; then a,=b,=43d,, and the only possibility is case (x). We can remove d; as before by a substitution (z+ Az, z). We may conveniently replace d,, d. which do not vanish, each by unity. The 6’s are now ur, w, (asx+by)w, 0, where either of a;, 6; may vanish, but not both, as we should then have a cone. Let us suppose a;+0, so that (modifying z) we can replace it also by unity. 104 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF The fundamental differential equation has the three independent integrals: a—y, w, U=2z2w—2ay+(1—);) y*. From these we can construct the quintic integral f= U {rv (a-y) + pw} + U (abedQa— y, w+ (a, Wc, d’, ef’ Va — y, wy =0. When w=0, this reduces to {— Qary + (1 — b,) yd (w— y) +4 [= Lay + (1 — bs) y%} (@—y) +a (w= yy | This to divide by ay’, leading to a=a’=0, A(1—b;)=0; but, if X=0, the surface degenerates, therefore b,= 1; the further condition that 2y=0, w=0 should be double gives b’=0, and the surface then reduces to S= (ay — zw)? (vw — y, w) + (ay — zw) («@ —y, WP wt (e@-—y, wPw=0 2... (XLI). The birational transformation ay —zw= ZW, «=X, y= Y, w= W converts it into 2(X —Y, Wi +Z(X—-Y, WP+(X-Y, Wy=0, a cubic cone, in general non-singular. The quintic has a triple pomt at e=y=w=0. The pencil of planes «—y+Aw=0 cuts the surfaces in two families of conics given by the equation of the plane and an equation of the form wy —zw+ pw*=0. We have now completed the case where «a= @=y=0, finding four new surfaces (XXXVITI)—(XLI), of which (XXXVIII) properly belongs to the case of three intersecting double lines treated in “Quintics I,” § 6. Since when a= 8 =y=0 the @’s all divide by w, all the surfaces obtained should belong to the class treated in “Quintics L,” § 10. On inspection we see at once that (XXXVIII) is a special case of (XVIII), (XXXIX) is a special case of (XXI) (the coefficients p, o of that equation being replaced by zero), (XL) is a special case of (XVII). If in (XLI) we write e—y=2X, «+y=2Z, z= W, w=Y we obtain a surface which is a special case of (XVI). The results of this section (IIIB), might have been obtained by assuming the result of “Quintics L,” § 10, and examining which of the surfaces there obtained had, or could have, double conics on them; but I have preferred to carry out the independent investigation of this paper, as a check on the work of the earlier paper. We have now completed the case when the double conic breaks up into two distinct straight lines. § 6. QUINTICS WITH A DOUBLE CONIC, CONSISTING OF TWO COINCIDENT STRAIGHT LINES, BUT WITH NO DISTINCT MULTIPLE POINT. The conic being 2?=0, w=0, the quintic can be written in the form f= wu, + uw + usw? + vyw* + vw! + cw® = 0. As at the beginning of § 5, we reduce the @’s to the form cut ae +acw+ dw, yut Ba+ Ow+dw, zut ya2+ Ow+dw, wv+tdw*, where 0;=a,+ b;y+¢;2(i= 2, 3) and wu, v are also linear functions of 2, y, 2. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 105 From the term independent of w in the fundamental differential equation we have as before 5uu, + 4anu, + 2° (oo +B EtY gE) =()\. Hence (i), if u, does not divide by x, we have 5u + 4ax=0, whence as before v = az, and du, du, diy _ : mre dy + dz ON aa Bs es to dy = Olsen eke ood soca (18), or (11), if % is a multiple of z, say a itself, we have 5u+4tax+ar=0; or u=—az. The secondary differential equation then gives v=2az. The coefficient of w* in the fundamental differential equation then gives mY d d\ (ee eh hae hae x P e: ALU: + # (a7. +8 dy ita) Us + Age Je tO gy + Ong + Beal atm + Sse =0. The second, third and fourth terms of this equation divide by 2°, so that avu, must do the same; but uw, cannot divide by # since =w=0 would then be a triple line, a case previously considered; hence a must vanish, and we revert to a special case of case (i). Thus we have only to consider case (i), in which the @’s are — ta? + a0°+a,0w+dw*, —4tary + Ba? + O.w+ dw, —4tax2 + ya? + Ow + dw, Larw + Fd’, subject to the conditions (18). Now u, must contain one or more terms of the form (y, 2)°, since otherwise =w=0 would be a triple line. By substitution in the differential equation we see that these terms are annihilated by the operator lea (boy + ¢22) 7 + (dsy + 52) ae (ihre By linear transformation of y and z we can always remove c,, and can further remove b, unless c;=b,. Applying the operator in this reduced form to (y, 2)? we must have one of the alternatives : 3b, +d,;=0, or 2b,+¢;+ d,= b, -—a,=0, or c;—a,=0, or 3c,+d,=0 ...... (19); if 6, is not removable we have the further condition b,=c,, whence: (GSS GS 1, Ch =8h OP Ch Sl OHO) =O cehcereceeasosacocece (20). We have seen (Case II. of § 4), that the constants d,, d,, d, can be removed by linear substitution if the three quadrics S,=6,—20/w=0, S,=6,—y6,/w=0, S,=0,— 20,/w=0 meet in a point not lying on w=0. In our case S,=— aa? +(aq,—4d,) cwt+dw*, S,=—ary+ Sa + a,rw (b, — $d,) yw + dw, S,=— aez+ ya? + asew + bsyw (cs—4d,) zw + dew. Vor. XX, Parr I. 14 106 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF Hence we can remove d, in two ways if a+0; supposing this reduction effected so that d,=0, the quadrics meet where z=0, (b.—4d,) y+ dw =0, by +(c; -—4d,)z+d,w =0, so that we can remove d, unless b,— 4d,=0, and then d, unless c,—$d,=0. If b,—4d,=0, we take the other factor of S,, viz, —az+(a,—4d,)w, and we see that we can remove d, unless a, —4d,=0 and d; unless c,—a,=0. We can proceed similarly if c,—4d,=0. Thus, if a+0, we can remove d, unless and we can remove d, unless It is now convenient to distinguish two cases according as 2 does not or does vanish. Case T: a+0. As we have just seen we can always remove d). Firstly, let us suppose that we can also remove d., ds. The coefficients of w® and w* in the fundamental differential equation give as before the alternative conditions a, + 2d,=0) orb; = 2d,— 0) onicy-20.—0 ord, —0) Aeeeeeneeeee (23). As before at least one term quadratic or cubic in z must occur in our quintic, Le. one of the terms a%z*w, xz*w*, yz*w*, zw, 22w* must exist. As before we deduce the alternative conditions 4b, + 3d,=0, or c;—a,=0, or c; —b.=0, or 3c;+d,=0, or 4c,+ 3d,=0...... (24). The only values of a, bo, c;, d, which satisfy (19), (23) and (24) are (a) 0, 0, 0, 0; (8) 1, 0, —1, 0; (y) 1, —1, 0, 0; of which we need not consider both (8) and (y), as they are obtained from one another by a mere interchange of y and z. If in case (8) we make the transformation ar—w=az", a, bs, c;, d, are respectively changed into 7, —4, 12; the constants d., d, reappear but can be removed again by a) 5? substitutions (y+ pw, y), (+ vw, 2); we can then apply the conditions (23), which are obviously not satisfied. Thus we have only to consider case (a): the 6s are now lau, —4axry+ Par? +anw, —taxz+ya°+(asv + by) w, Sanw. We can further remove £ by a substitution (y+, y), y by a substitution (z+ va, 2), and then a; by a substitution (2+ Aw, z); it is convenient further to remove a, by a substitution (y+ aw, y), thereby reintroducing d;. The 6's are now tag, —Zary, —tarz+byw+dyu*, Sacw. The fundamental differential equation has the three independent integrals ay, yur, aU, where U =aaz — byw —idyw*. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 107 From these we can construct the quintic Fa Ys yt O(a, eye) ts UB tb Ones nso caemecen ace em sce (XLII). This is the most general quintic integral if U does not factorize, but if 6; = d;=0 we have an integral of a more general form “(y, zi+ae(y, zPw+ty, zPw?=0, which has a triple pot at e=y=z=0, contrary to our present hypothesis. The transformation «=X, y= Y, w= W, U=YZ converts (XLII) into (X2, YW)?+ XZ (X*, YW) + YAW =0. This is a slightly specialized form of the quartic (A) (“ Quartics,” p. 343), which is known to possess an integral of the first kind. The quintice surface has a triple point at e=y=w=0. Secondly, let us suppose that after d, has been removed, either or both of d, and d, cannot be removed by substitutions (y+)w, y), (2+ mw, z). If d, cannot be removed, we have seen that the conditions (21) must hold, so that either (a) a,=1, b,.=1, c;-=—4, d,=2, or (8) a, bo, ¢s, dy all vanish. In case (a) since the conditions (20) are not satisfied, b,=0; the terms highest in z are annihilated by d d d d ler? + a —4axr a2 ; aw? 4ar , 6 yy 2 : (lar + aw) 7 +(¢ tany + Ba Sie am ang 2) a tanz —4zw) dg + (etrw + wi) G The terms highest in z must be either z*w*, or one of a°z*w, wz*w*, yz*w*, 2°w*; on substitution and arranging by powers of w we see that none of these terms can exist; so that this case fails. Since },=0 there is in this case no essential difference between y and z; so that the same reasoning shews that if d; is not removable, the case corresponding to (a), viz. that in which a,=c,=1, b,=— 4, d,=2, fails also. Hence if either d, or d, or both cannot be removed we have only to consider case (8) in which a, b,, ¢;, d, all vanish. The @’s are then: taa®, —4ary+ Bo? +anw+dw*, —taxz+-ya+ (av + by)w+dw, Saxw. By substitutions of the form (y+) w, y), (y+As#, y), (2 +Asw, 2), (2+, Zz) we can successively remove a>, 8, d3, y; finally by a substitution (z+ my, z) we can remove ds, unless d, vanishes. The 6’s are now: laa, —4taryt+dw*, —4taxz+ byw+dw, Lac, where one of d,,d; vanishes and the other does not. These two cases need only be distinguished it (be-108 Case (i). d,=0, d;+0. The fundamental differential equation is the same as the form to which it was finally reduced in the preceding case, viz. when d,, d; were removable, but it was found convenient to reintroduce d;. The integral is accordingly (XLII), as before 14—2 108 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF Case (ii). d.+0, d;=0, 6,+0. The terms cubic in y, z which must exist in uw, are annihilated by b, = so that z they reduce to a single term Ay*, where +0. The coefficient of w° in the fundamental differential equation, gives ad dy Gar + by “ +d, —v=0; hence v, contains no term of the form 2/fyz. 2 y The coefficient of wi now gives pp2 d » \ ay f d d — (a FR 4a) v, + bey Gest eg = ()) The only term in this equation containing y* is 3Ad,, which must accordingly vanish, contrary to hypothesis. This completes the whole case when a+ 0. Case IIT: a=0. First, let us suppose that d,, d,, d; can be removed; then the analysis employed in the corresponding sub-division of Case I. (2+0), shews that the constants a,, bs, ¢3, dy either (a) all vanish or (8) have the values 1, 0, —1, 0. In case (a), the 6’s are 0, Pa+aanw, ya? +(ase+ by) w, 0. The fundamental differential equation has the three independent integrals 2, w, U=ary + (ayvy + bbyy?) w — (Ba? + agrw) z. If U does not divide by a function of # and w any quintic function made up of these integrals can only contain U linearly, so that z also only occurs linearly, a case of failure. G) Jf b;=0, U divides by a and can be replaced by the simpler integral V=yey + ayw — Buz — azw; if V cannot be further reduced by the removal of a factor; from #, w, V we can construct no term cubic in y, 2 conjointly; so that we revert to the case of a triple line ; if a further factor linear in « and w divides out, then V reduces to a linear function of y, z say Xy +z, and then our quintic is a function of a, w, Ay + pz, 1.e. a cone. (u) Jf b;+0, but B=y=0, U divides by w and can be replaced by the simpler integral T= ayry + bby? — a.xz; but from x, w, W we can construct no quintic containing terms of the type (y, 2)° w*; so that we have a failure as before. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 109 In case (8) we can remove az, ds, b; in the usual way, so that the 6’s are: zw, Bx, yx—zw, 0. The fundamental differential equation has now the three independent integrals w, U=Ba?-2yw, V=x(ya* —32w). From these we can construct b = (B°V2 — fF U*)/w = 6By24 (yy — Bz) + 9Ba22w — 128oPa%yPw + ByPyw. The most general quintic integral that can be constructed is f=a6 +bUV 4+ cU*w +dVu? + eUw* + cw? = 0... e cee es (XLII), where a, b, c, d, e, « are arbitrary cvefticients. The birational transformation z= X, w= W, 2yw — Ba? = YW, x (ya* —32zw) = ZW? converts the surface into a(S2W—Y*)+bYZW+cYW+dY W?+eZW?+%«W*=0, a cubic cone, which is in general non-singular. The quintic has a triple point at s=y=w=0. Secondly, let us suppose that d, can be removed but that one or both of d,, d; cannot be removed; suppose first that d, cannot be removed; then 6,—}d,=0. Combining this with conditions (19) we have five possible sets of values for a,, bz, cs, dy; viz. 1, 0, ae ifs 0 (2), le if or 4, 2 (11), 3, —2, +3, —4 (111), ifs = 3, 2, a 6 (iy), 0, 0, 0, 0 (v). In cases (i)—(iv) b, vanishes, and we can remove d, by a substitution (z+ dw, 2). The conditions for the existence of a term quadratic or cubic in z readily shew that cases (i) and (ii) are impossible; the corresponding conditions for y shew that case (iv) is impossible. In case (iii) the terms independent of # in the quintic satisfy the equation f ~~ d d ) 9) ay —= Df NP Ks ce Ae ne eB a ria Qu? duh? 0. This equation has the two independent integrals z*w*, 2y/w+d,logw; from these no quintic terms of the type (yz)*w* can be constructed. Thus cases (i)—(iv) all fail. In all these cases 6, vanishes, so that there is no essential difference between y and z; thus if we had assumed d, not removable we should have found four cases of failure exactly corresponding to (1)—(iv). Hence whether it be d, or d; that is irremovable we have only to consider case (v), in which a, b., c;, d, all vanish. The 6’s are now: 0, Bet+aaw+dw, ya? + (asx + by) w+ dw, 0. 110 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF The fundamental differential equation has the three integrals xv, w, Uz=yaty+ ascyw t+ bby w + dw y — (Ba + agew + diw*) 2. In order that our quintic should contain terms cubic in y, z conjointly U must occur raised to the power 2 at least; hence it must divide out by a factor depending only on «, w. Tf b,+0, the only factor which can divide out is w; and then 8 =y=0 and U can be replaced by the simpler integral V=a,ry+ $b,y?+d,w*y — a,vz—dew; now V* contains 7}, which cannot occur in our quintic; hence V still cannot occur raised to the second or higher power; and we have a case of failure. If 6,=0, U is linear in y, z; hence in order that terms cubic in y, 2 conjointly should occur, U* must occur; hence U must divide by a quadratic factor depending only on « and w, and the residual factor is of the form V = ry + pz, where X and p are constants. Our quintic is accordingly a function of three variables only, viz. x, Ww, Ay +z, and is a cone. This completes the case when d, can be removed. Thirdly, let us suppose that we cannot remove d,: then a,=4d,; combining this with the conditions (19) we have for the fundamental constants five possible sets of values, viz. : Se th ie 6 (4), Sh =, 6 (i), il 1, -—4, 2 (iii), 1 2A ot Le (iv), Om 50" 00 ed) (v). In all cases except the last b;=0, so that there is no essential distinction between y and z; hence it is unnecessary to consider cases (11) and (iv), if we consider cases (i), (iii), and (v). Case (1). We can remove in the usual way d,, d;, ad, a3, reducing the 6’s to Bew+dyw*, Ba®*—2yw, ya*—Tzw, 3w. It can then at once be verified, as on previous occasions, that no term quadratic or cubic in 2 can occur in our quintic. Case (111). We can remove d, by a substitution (y + Az, y), as by a substitution (2+ pa, 2) and then d, by a substitution (z + vw, z); so that the 6’s are cw+dw*, Bar+acwt+ yw, ya®?—4eu, wr As before no term quadratic or cubic in z can exist. In case (v), we can remove d,, d; by substitutions (y+ A, y), (2+ ma, z); so that the @’s reduce to: dw*, Bu+a-ew, yat+(ase+by)w, 0. The fundamental differential equation now has the three independent integrals w, 48at+ha.e*w—dyyw*, 4Bb,24 + 4 (ab; — yd,) aw — (hax + byy) ew + djztw. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 111 From these we can construct no integrals quadratic in y or z unless at least one of the last two integrals divides by w, so reducing. This can only happen if 8@=0, or b,=0. If b,=0 we can remove 8 from the differential equation and therefore from its integrals by a substitution (y+ Az, y); hence whether 6; vanishes or not, 8 must vanish. We now have the simpler integrals w, Uz=aga?—-2dyw, V=2 (ab; — yd,) x — 3a,d,a°w — 6b.d,xyw + 6d,2zw. From these we can construct the quintic integral ¢=(AU*+,V*)/w, where X and wu are constants so chosen that days + 4 (asb, = yd,)° =(/)) The most general quintic is now FH= OG Wi Ve eV HO iage dan sexesseees ote (XLIV), where ,, «. are arbitrary coefficients. The birational transformation U = YW, V =ZW?2, 2 =X, w= W transforms the surface into V8 4+ p2W+(Y, WP W4+ «ZW? + «.YZW = 0, a cubic cone which is in general non-singular. The form of the quintic requires some modification if a,=y = 0, as in that case the integral ¢ which we have used becomes indeterminate. The simplest integrals are now y, W, asx? + 2b,ry — 2d,zw. If from these we construct the most general quintic with z*=0, w=0 double on it we find that w must be a factor of the whole expression, unless b;=0, If 6,=0, a; cannot vanish or we should have a cone; we may conveniently take each of d,, a; to be unity and the quintie now becomes f= — zw) (y, w)! + (@ — 22w) (y, wP w+ (y, wy w?=0 ......... (XLV), the 6’s being now w’*, 0, zw, 0. If we use the transformation 2*— 2z2w=ZW, «= X, y= Y, w= W, we get the cubic cone Z(Y, WY +Z(Y, WyP+(¥, WP=0, which is in general non-singular. This completes the case of a=0, so also the whole case when the double conic degenerates into two coincident straight limes; we have found four surfaces represented by equations (XLII)—(XLV). 112 Mr BERRY, ON CERTAIN QUINTIC SURFACES. § 7. CONCLUSION. We have now given the equations of a number of quintics satisfying the conditions of our problem as defined in the introduction. Each surface possesses at least one integral of the first kind of a total differential, and has a double conic but no further double line; cones have not been taken into account. According to the system of classification adopted there are twenty-three surfaces, the equations of which are marked by Roman numbers in continuation of the enumeration given in “Quintics I.” Three of these surfaces possess a triple point not lying on the double conic ; they are Cayley’s quintic with a double lie and two triple points, and two limiting forms of it; they are represented by equations (XXIII) (with a modified form) and (XXIV). Seven surfaces represented by equations (XXV)—(XXXI) have a double point not lying on the conic. The remaining thirteen surfaces have no double or multiple point distinct from the double conic; in the case of jive of these thirteen surfaces, represented by the equations (XX XIT)\—(XX XVI), the conic is non-degenerate, in the case of four surfaces, represented by equations (XXX VII) and (XX XIX)—(XLI), the conie degenerates into two intersecting straight limes which are distinct, while in the remaining four cases, represented by equations (XLII)— (XLV), the conic degenerates into two coincident straight lines. The surface (XXX VIIT) should have been given in “ Quinties I,” § 6, and only differs slightly from the surface (VII) of that article. Six of the twenty-three surfaces belong to a class considered in “ Quintics I,” Part U1, viz. : those for which the fundamental quadrics 6 have a common linear factor; three, viz. (XXY), (XXXII) and (XL), are special cases of (XVII), two, viz. (XLI) and (XLV), are special cases of (XVI), while (XX XIX) is a special case of (X XI). The surfaces of this paper share with all other known quartic and quintic surfaces, which admit of integrals of the first kind, the property that they can be transformed birationally into cones; so that in all cases the geometrical genus (p,) 1s zero and the numerical genus (p,) is negative. All the surfaces of this paper can be transformed into cubic cones, so that p, =—1. Each surface possesses a family of rational curves corresponding to the generators of the cone. I have not in general worked out the equations of these curves, but they can be immediately deduced from the transformations, which I have always given, by means of which the surface can be transformed into cones either directly or via quartics. None of the surfaces of this paper are ruled, a result which is in agreement with Schwarz’s enumeration * of ruled quintics. * «Ueber die geradlinigen Flichen fiinften Grades”: Crelle’s Journal, t. 67 (1866); reprinted in Gesammelte Abhandlungen, t. 2, pp. 25—64. V. The Law of Error. Part II. By Proressor Epcewortu. ANALYSIS OF CONTENTS. The first four attributes of the typical case (defined Part I. p, 38) are replaced by more general ts conditions A : ; : : : : 3 : - : ; : ; : ; - 113—115 The condition that the range of the element’s frequency-locus should be finite is not necessary for the new proof; is in general fulfilled sufficiently for the old proofs. ; : : : 115 The law of error extends to the case of elements with several dimensions. < : : - 116—120 The law also extends to the case in which the compound quantity is a function other than linear of the elements, capable of being expanded in ascending powers thereof : : . 120—126 Analogously in the case of elements which are slightly interdependent the law of error is fulfilled; there is no other general form for the frequency-locus of a quantity which is the sum (or @ fortiori a non-linear function) of elements which do not vary independently . - - 126—7 The relaxation of the last three typical attributes further widens the domain of the law of error 127—130 AppFNDIX ‘ - 5 e A : : : . A ‘ : : : 5 é cs 131 THE conclusions which have been proved for the “typical” case are now to be extended to more general cases. These may be reached by removing or relaxing the conditions which form the attributes of the typical case*; beginning with the one which was last in the original enumeration. Removal of the eleventh condition. The condition that the finite difference A£(= Az), being the interval between adjacent values of an element, shall be the same for all the elements may be dispensed with by taking Aw as the greatest common measure of all the intervals (between adjacent values) presented by all the elements (in the long run). Thus if an element which (measured from its lower extremity) presents the values ON Aa Aaa Ne is to be combined with a binomial element which presents only the values 0 and vAa, it is allowable to regard the latter element as presenting the values Az, 2Az,... (v—1) Az with zero frequency. * See Part I. p. 38, Vou. XX. Parr I. 15 114 Pror. EDGEWORTH, THE LAW OF ERROR. Substitution for the tenth, ninth, and eighth conditions. The three conditions which (in the reverse order) come next may be reduced to one: namely that a descending series should be formed by the coefticients hk, hy, k....k,*, each divided by k (or 2k) raised to a corresponding power, viz., respectively $(0+2), $(1+ 2), 4(2+ 2),...4(¢+2). For thereby it will be secured that the mean powers of the compound w will be broken up into groups forming successive approximations. Thus + 2p ! kp —1) k,\ op) = 2P (IOSD: me ple aera ee get) — 2p +1 oa f uy — 2 (ge) hs) st hetero p—l29 |3:k SiO a et Also the expression for the ordinate} can be exhibited as a descending series, viz., Ide ye _ 1 dty, eg! d'y, Yas a 3! dx?’ Ks 4! dx# ahs suede aie where x=a/V2k, k,=k,/(2kye, 1 : d’y, 1 ay i xe . = (2k) ; Yo TE -e-**, and accordingly as = (2k) aE The convergency of the series is thus secured, provided that # is not too large with respect to 2h§. This convergence tends to be impaired by the inequality between the elements in respect of mean deviation against which the tenth condition is directed. This may be seen by considering the case in which some of the component elements have a mean- square-of-deviation which is very small in comparison with the corresponding coefficient for the other elements. Thus suppose the system of elements to consist of two classes, m, elements assuming the values 0,1, 2, 3,...18 with the same frequency as the number of pips on a domino taken at random, and m, elements assuming the values 0 or 1 according as a tossed coin turns up tail or head. We have for & the mean-square-of- deviation for the compound m, x 16°5 + m,x 0°25. For k, we have m,x — 83°325 + mx — 0125. Unless m, is very great in comparison with m, it will not contribute much to the con- dition that h,/k? should be small. It must be supposed that the elements are not|| regular, otherwise the inequality of their dispersion would not impair the regularity of the compound. Another circumstance unfavourable to the fulfilment of the essential condition is extreme asymmetry on the part of the elementary locus; which is guarded against by the eighth condition. The effect of asymmetry is illustrated by the case of the binomial when one of the alternatives is very improbable. The quantity mpq which affects all the coefficients k,, k,,... as a denominator then tends to be small, and accordingly those coefficients tend to be large‘. Analogously in general when an element has a very * See Part I. p. 44. . applied to the law of error as generalised in this paper ; + Ibid. p. 43. t Ibid. p. 45. the term ‘‘normal” being confined to the first approxi- § See Appendix, Note 2. mation. || It will be remembered that the term “regular” is §] See Part I. Sect. 5. Pror. EDGEWORTH, THE LAW OF ERROR. 115 asymmetrical locus its mean powers will be small, say ey?, ey’, ey, &c., where ¢, &, &” are small fractions of the same order. Accordingly the k coefficients will not form a de- scending series (unless m is very large). E.g. k,= = (e’y'— 3e*y*)/4e%y'm?; where ¢€ and € are presumably of the same small order. In general, whatever the abnormality of the elements, whether it be due to asymmetry or other incidents, and whether or not it be aggravated by inequality between the elements in respect of dispersion, the regular law of error will be generated, provided that the number of the elements is sufticiently great. The number need not be great when the elements are nearly regular (themselves perhaps compounds). If they are perfectly regular the sum of any two elements will obey the regular law of error. Removal of the seventh condition. In what precedes it has been taken for granted that the mean powers of the elements are finite; as follows from the attribute that the range 1s finite. This attribute is only required in order to afford that inference; if the method of proof recommended in Sect. 1 of Part I. is adopted. That proof is not affected by the range of the elements becoming infinite, provided that their mean powers remain finite. But in employing the method originated by Professor Morgan Crofton, in cases where the range is infinite we are met with a difficulty when we proceed to expand the expression for the frequency of the compound in terms of the mean powers of the elements*. Some of the constituents of these means now become infinite; and accordingly we are employing Taylor's theorem for the expansion of f(#+/) in ascending powers of h, in cases where his infinite. A similar difficulty is presented by Laplace’s method when it is attempted to expand x(a) (in our notation) in ascending mean powers of an element+. As Laplace himself says, “si les erreurs peuvent s’étendre a l’infini, l’application de la méthode précé- dente peut offrir quelques difticultés?.” These difficulties may be removed by the following theorem. An element with indefinitely great range may be reduced, without sensible effect on the locus of the com- pound, to an element with range finite and small in comparison with the “standard deviation”§ of the compound; provided that the mean powers (up to the tth) of deviations in excess of the centre of gravity, and likewise the mean powers of deviations in defect— each set being taken in absolute quantity—are finite and not very great, so that the tth root of the mean tth power is a small quantity|| relatively to the standard deviation of the compound. For consider a case in which the area of the element above its centre of gravity is p, and below the centre of gravity q(p+q=1). Let OP, measured from the centre of gravity to the right, represent the tth root of the mean ¢th power of deviations in excess ; and likewise OQ the tth root of the mean ¢th power of deviations in defect. And let us endeayour to arrange that as large as possible a portion of the element should lie above a limit considerably greater than OP, say 30P; and likewise as large as possible a proportion below 30Q. The arrangement most favourable to this result would be that the portions of the area not outside these limits should be huddled up in the neighbourhood of the centre * See Part I. Sect. 2. quantity here denoted by ./k. + See Part I. p. 52. || A quantity of the order 1/,/m according to the scale + Théorie analytique des Probabilités, Bk u.ch. 1v.§ 18. of units recommended in this paper. See Part I. Sect. 1, § Professor K, Pearson’s convenient term for the and Appendix, Note 1. 15—2 116 Pror. EDGEWORTH, THE LAW OF ERROR, of gravity so as not to contribute to the mean powers of the element. Let pp be the proportion of the area outside 30P; and likewise p’q the proportion outside 30Q. If all the portion outside 3OP were indefinitely near that limit, pp(30P)"=pOP". But if the portion outside 83O0P is more dispersed, p . OP= pp (OP’)", where OP’ is greater than 30P. Accordingly p+(4)". Likewise it may be shown that p’ is very small. A fortiori when the element has a less peculiar form of frequency-locus than that which has been supposed*. Therefore (it being granted that the mean powers of the elements are finite) all the elements may be considered as ranging within limits sufficiently small to permit of the expansions practised in the methods originated by Laplace and Professor Morgan Crofton. The pro- portion of observations involving values of elements outside those limits must be so small as to be insensible when distributed over an area (bounded by the frequency-locus of the observations compounded of elements) which is nearly equal to unity. Removal of the sixth condition. Beginning with the case of two dimensions, let us represent the frequency-locus of the rth element by the equation ¢, = ¢$,(&,); signifying that the probability, or comparative frequency in the long run, of the concurrence of the values & and » is for the rth element $,(&,7) AEAn. Let the pair of values contributed by the rth element to the sth observation be (,&,., .7,). The sth observation consists of a pair of values for the compound quantity (,”, sy) such that rT=™ r i : = w= > £5 Y= r=. 7 Beginning with the method placed first in our First Part+ we have to find a repre- sentative surface whose mean powers of deviation from the centre of gravity are approxi- mately equal to those of the actual locus; not only mean powers for one of the variables, such as the mean of a? or of y2, but also the mean of products such as ax y’. The latter mean may be designated thus (#, y)%. If f(y) is a surface approximating in form to the actual locus [/fx'y"f (xy) dewdy|—the square brackets denoting integration between extreme limits— =(a, y)%), Take the centre of gravity as before for the origin, and transform to principal axes by putting «=X cos?—YVsin @#, y=Xsin@+ Ycos6, where @ is taken so that (XY, Y)*)”, the mean product of the coordinates pertaining to the compound, may vanish. This will be effected if tan 20=2 (a, y)%/(@, y)°— (a, y)»*t. Following the analogy of the simpler case§ let us express the generating function of il = er : ; = : nt! (X, Y)® © in a form suited to our purpose. The generating function of 1 ap TH ada is es*+P8Y: the coefficient of a‘8" in the expansion of this function being a Xt Ves * See an example in Appendix, Note 5, + Section 1. + Compare ‘‘ The Compound Probability Curve,” Phil. Mag. 1896, vol. x1. p, 208, by the present writer. § Part I. p, 42. Pror. EDGEWORTH, THE LAW OF ERROR. BW And et*+8¥= es*=+82H (if =, H, are the coordinates for the rth element referred to the principal axes X and Y) =(1+a=,+ 8H,) + 5 (OE + 2a8=,H, + 6°H,2)+... = 1 = x (1+ a. + BH) + 55 (RY + Hehssor cree tonec y+... = 1 sore x (1+4E n+ S8Hm)+ 55 (Ode gies teieaeecsceeces Veereee Therefore, taking averages on both sides of the identity, 1+aX¥%+ BV +4(2X + 2aB(X, VY) + Beye) +... is identical with the mean of the last written product; that is, as the elements vary independently of each other 1 1 en +8H\+ 21 (a?,, oX1+2a8,, x, +B%, 21) +3) (3, ox 0° +3y, 1x, 078) +... 1 oe 4 1 gone ay] (272, oka +2a8,, x. +B", 2) 3] (s, oXa+-..)+... . 2 Ry: 1 ; : where ,,,*, 1s the coefficient of ie! a?8? in the expansion of log ( ihe a=, + BH,” 25 = (a7=,°) 2a8 (&,H,)%: + 67H,” Seer el \ : r=m r=m Put ..M, or more shortly K, for = . «,", 42K, or more shortly K', for = 4 .«,”; and r=1 r=1 a : : 4 Le generally ,.K a p,qkr; and we obtain for the generating function of aval (xX, VY) remembering that X”, Y, (X, Y)?® all vanish— 1 : ae ae a (a2K+82K ')+ 3 (s,0Ka°+3,, .Ka%8+3,,Kaf? +), 3K6%)-+... 3 where ,,X is of an order inversely proportional to the (p+gq—2)th power of Vm (if the elements are supposed to be typical*, except in that they are two-dimensioned). The values of (X, Y)®") thus given may be expressed as integrals of a repre- sentative surface, if we convert the generating function into an operator by omitting the first pair of terms in the index, viz. $Ka*+4K’g, substituting in the remaining terms for and putting for operand the (normal) function z Ex =e + as) on VKR’ \2K * 2K’) For it will be found that the expression given by the generating function (X, Y yo, (ty a, — and for £8, — say Z. Ce! * It is often convenient in enouncing the relaxation of one typical attribute to treat the others as in force, where no new difficulty is introduced by their relaxation. 118 Pror. EDGEWORTH, THE LAW OF ERROR. is identical with =| _{_ xwezaxay; where Z is the function which results from the performance on Z, of the operation which has been indicated. For by analogy with the case of one dimension* it will be found that any term in one of the compared expressions, e.g. a term which involves A” and K “4, is equal to a corresponding term in the other expression. The comparison may be effected by the equations i i Ye» YZ, = 2p! 29! KeK’: 2Pp! 229! ae he a ikae sees ges iia) ROT: ti¢#! ec es a ei De Ih)y dXdY= a where t=2p+r, t=2q+s. It remains to transform the expression which has been found back to the original coordinates. This is effected by substituting in the operand Z, for X and Y their values in terms of z, y, and @, for K and X’ their values in terms of (a, y)®), (a, y)@, (v, y)®®, say k, l, and k’ respectively, and for tan @ also its value in terms of those coefficients. There results for the operand, and as the first approximation to the required law of error, 1 2) = ——_———— Exp — (k'a? — 2lay + ky’)/2 (kk — E). Ia N kk — *P a J Y) ‘ ) For the operator it suffices to change each coefficient of the type ,, to the corre- sponding symbol in lower case, viz. »,,k where »,,k has the same relation to z that , has to Z; that is, »,k is the difference between (a, y)” as it actually is, and as it would be if z,,,-. the (p+q—1)th approximation were correct+. This may be shown by considering the group of terms which are of the order (p+q—2)/Vmt, as having the character of an emanant. For instance, in the case of p+q=3 the group (of the order 1/Vm) is xs + 82x Teagy t+ Buck aygpt ok ays 3! Filaek Set ; d'Z, aZ, oA): This expression is the mean, that is, the quotient of a sum, of constituents each of which is of the form i Ly seca ORE } ale sé 3X"} 3 7 3! dX* dXedY where X’Y’ is a particular pair of values assumed by the compound quantity. But by the theory of emanants§ this constituent is identical with 1 f zy Q,,/2,,/ dz, \ xv Ot ent alate 3! da J dard Yy fh * Part I. p. 44. + On the supposition of typical—not in general regular + Or the sum of the corresponding differences for the | —elements. elements. Cp. Part I. p. 44. § Salmon’s Higher Algebra, Art. 125. Pror. EDGEWORTH, THE LAW OF ERROR. 119 where «’y is that pair of coordinates referred to the original axes which corresponds to X’Y’. Whence it follows that the group AZ, 301K ot... } ofc Stra | gl is identical with the group Like reasoning may be applied to any other group. With the aid of premisses furnished by the foregoing proof of the law of error in two dimensions, the same conclusion may be obtained by the proof which is based on partial differential equations*. When a new element is taken in we have now for the first approximation : - dz (1) 02 = [S88F}u (En) MEM) 95: Ez (2) éz=[SSEndm (E, 7) AEAn] Indy (3)- dz =4[SSq°¢,, (E, 7) AEA] ae the square brackets indicating that the summation extends over the whole range of the new element. By first principles+ the three bracketed quantities may be replaced respec- tively by Ak, Al, Ak’, the additions made by taking in the new element to fh, l, k’ (respectively representing 2”, (x, y)">”, y®). We have thus the system of partial differ- ential equations: dz l1dz dz dz Eds eelraez => 7) ees (3) =7=5 78° 2 dx dl dudy dk 2dy If we transform to principal axes / disappears. There must be satisfied the two partial differential equations Wiel dz _1ee dK 2 ¢X*° COO: sk a For subsidiary equations we obtain, by parity with Professor Crofton’s reasoningt in the case of a single dimension, (1) (2) dZ dZ . 9 Ags (I) Z+xX ax + 2k qk 0; P dZ ,dZ Combining the subsidiary with the leading system, we obtain the first integrals: dZ Le where C, does not involve X, nor @,, Y. Further, as by the fundamental theory the odd * See Part I. Sect. 2. + Cp. Part. I. p. 46. + Encyclopedia Britannica, 9th edition, Art. ‘‘Probability,” p. 781. 120 Pror. EDGEWORTH, THE LAW OF ERROR. powers of X and Y in Z may be neglected in a first approximation*, when X =0, = =0, and therefore C,=0. By parity of reasoning C,=0. Whence by a second inte- gration—utilising the condition | / ZdX dY = 1—we obtain the normal surface. ms ~~ There is no difficulty in extending this method to further approximations on the lines indicated by the present writer in a former papert; with the aid of the fundamental theory stated in this paper. The proof of the law of error for two dimensions on the lines traced by Laplace for one dimension} presents no additional difficulty; so short is the step from the generating function (or the operator) which has been above written to that function in a for one dimension§, in a and 8 for two dimensions, which is proper to Laplace’s method. Nor does the use of the method which is based on the condition of reproductivity present peculiar difficulty in the case of two dimensions]. Nor need we be detained by the extension of the theory to dimensions more than two; nor by relaxations of the typical conditions, such as have been already considered with reference to one dimension. Slight relaxation of the fifth condition. So far we have supposed the method of com- pounding the elements to be of the simplest sort, namely, summation. To this species may be reduced the case in which the compound is the sum, not indeed of the m elements, but of m magnitudes each of which is a given function of a different element, say n= Van (&) + Wr. (&) +... + Vn (Em). Here if £ obeys the law of frequency »=¢,(&;), the statistical quantity which we may call [y,] obeys the law dy, 7 = $(&,) ae where for &, is to be substituted its value in terms of [y,] obtained from the equation [Wr] = vr (&). The compound quantity is thus shown to be the sum of m elements [vi], [ye], --- [Winds each of which fluctuates independently. Accordingly (af the new elements satisfy the essential conditions) the law of error will be set up. This variety includes the important case in which the function whereby the elements are compounded is linear. Functions other than linear are not zn general amenable to any of the proofs which have been given, Nor is there any universal form towards which the frequency-locus of such aggregates converges. If we are free to employ any combination of the elements we * See Part I. p. 47. variety is where y (¢)=s*, a being a positive integer; the + Phil. Mag., 1896, vol. xu1. p. 208, case discussed by Todhunter, History of Probabilities, Art. + See Mr Burbury’s extension of Laplace’s method, in 1006. Among known and simple cases of relations other Phil. Mag. 1894. than summation by which the law of error is set up may be § See Part I. p. 52. mentioned here the median and percentiles of a large set of || See Appendix, Note 6, elements ; though they cannot be classed as functions of | Another important simple case belonging to this the elements. Pror. EDGEWORTH, THE LAW OF ERROR. 121 can arrange that any assigned function shall be the limiting form for the frequency-locus of that combination when the number of the elements is increased without limit. For consider the frequency-locus of a statistical quantity, #’, which is some function of the sort of compound which we have hitherto been considering, viz. a sum of m independent elements; say « = (a), where e=£&,+&+...+€&,*. Then, if @(x) denote the normal error-function, da: da rived from the equation z’=¢(x), The resulting expression in «’ is to be identified with ' we have for the frequency-locus of «', @(«) if for # is substituted its value in 2’ de- some assigned function, say f(z’). We have thus @ (2) oe =f(2’). Whence O(2)+C=F(z2’), if capital letters are used to denote the respective indefinite integrals. Whence zw = F~ (C+ 0(2)). For example, let f(#) be (for positive values of «)(a—«’)/a. Then F (a') = (aa' — $a")/@; which is to be equated to V+ @(z). Thus w’? — 2az! + 2a2(C + @(x)) = 0, ? Satie Vi =a Lea) Remarking that © is a quantity, which mereases from zero to 4, as w increases from zero to infinity, let us arrange that « should increase from 0 to a while « increases from zero to infinity. This is effected by taking the lower, the negative, sign in the expression for «’, and putting C=0. The resulting frequency-locus for positive values of « will therefore be of the prescribed form y=(a—a’)/a’, if «’=a(1—V1—20(a))+. By a proper change of signs the result becomes applicable to negative values of # and «2. The larger the number of elements the more closely will the actual frequency-locus of « approximate to the right line (corresponding to the assigned function) as a limiting form. A fortiori there is no unique limiting form to be expected when the compound involves the elements, not implicitly as a function of their sum, but in any other way. Of course if the function can be expanded in ascending powers of the elements which after the first may be neglected, the case is reduced to that of a linear function. Between this simple case and the most general one—both of which have been now considered— occurs an intermediate, probably extensive, case: where the compound is expanded? in * The writer has employed the form x’=ax+bx?+ex? by Dr Burgess in the Trans. Roy. Soc. of Edinburgh, for the representation of abnormal groups (Journal of the vol. xxxrx. 1900. Statistical Society, 1900). + Quantities of the order (1/,/m)* in relation to the first + Laplace has expressed 0 (.c) in several forms; discussed term of the compound magnitude being retained. Vou. XX. Parr I. 16 122 Pror. EDGEWORTH, THE LAW OF ERROR. ascending powers (and products) of the elements, which are not neglected, but retained down to a certain order of magnitude, say the tth. It may be enquired whether the regular law of error is applicable to such expansible functions of elements. The answer to this question may be found by showing that there is a generating function for the mean powers of such a combination of elements essentially resembling the generating function which has been found for the mean powers of a compound formed by a sum of elements*. The following method of determining the law of frequency for expansible functions has the advantage of affording a fresh proof of the law of error for sums of elements. If 2’ is the given function of the elements, and y’ its sought law of frequency, y’Az may be considered as a certain strip of the locus w= d, (&) x om (&) ams Pin (Em) »which (multiplied by A€,A&,... A€,,) represents the probability that any particular system of values for the m elements should concur (y=@, being the law of frequency for the rth element). The slice of this “solid” which contains all the “points,” or systems of £,, &...&,, for which the given combination of these variables is between? 2 and «+ Az’, is equal to y’Axv. Change the system of independent variables from §&,, &, &... &, to x’, &, &...&,. The required strip may be written [//... Av wd&.dé,... dE]; where the integrations are to extend between the extreme limits of the variables: ,w is what w becomes when for &, is substituted its value in 2’, &, &..., and this form of w is multiplied by (the determinant proper to a change in the system of variables, here) the differential dé, da’ ” of that value of &, with respect to 2’, The handling of this integral is facilitated by two propositions which have been above established. (a) The equation for the locus of any element may be put, without loss of generality, in the form Ko = 1 2a, 1 1 aye 3 aR rors hak — Geyi 3 (12E/V2« — 8&/(2«)?) + (xy 4 ) (12 — wee) + ma the form of the regular law of error}, where «, «,, «2, &c. are determined in the usual way by the mean powers of the element under consideration. For the mean powers of the element as determined from the proposed locus are the same (up to any assigned power) as the given mean powers; and it has been postulated that when one locus has the same mean powers as another they may be considered as identical§. (8) The portion of any element which lies outside a limit equateable to « multiplied by a small numerical * See Appendix, Note 7. + In the form, but not in general presenting the essential + Or rather at x’, if the function designated by x’ varies _ character of the law of error, viz. a series of approximating with its variables £,, &, ... discontinuously by very small terms. steps. § Part I. p. 40. Pror. EDGEWORTH, THE LAW OF ERROR. 123 coefficient (small with reference to /m) is an inconsiderable fraction*. By these propo- sitions w, the locus of concurrent elements, may be put in the form+ ieee Sl ley 2a 4 A 1 a= —F 2/2) Ky — $o7/2K0.. = rel =a es ae Ss — Ae : Q é 2 E (2,%)8 3 i se] , ate =/Ke 4! (12 ate a) ? where the portion within square brackets outside the exponential breaks up into groups forming a descending series (the variables, by proposition (8), being restricted to the order of the corresponding power of ;«,1). These theorems are now to be employed in cases where «’, the given function of the elements, may be expanded in ascending powers thereof, say a= a,€, ar an&s +... On&n, fs Buk ate 2B2FE. ar 2BiskiEs ar andes) ar wnb? + Byker + 61206: E2Es + tee where the a's, §’s, and y's are respectively of the order 1, 1/m, 1/m..., as in general (with respect to the typical case§) it will be found necessary to postulate. Let us begin with the simple case in which the f’s, y's, &c. vanish, and the a’s are all equal to unity, the case already treated by other methods, in which the compound 2’ is equal to the simple sum of the elements. By the present method we have for the locus of a’, when in w (as above directed) there is substituted for £ its value —&—&,...—&,+2', say —&4R8, (i being in this case unity), an expression which may be put in the form 7 a I a ae ee ne Q wee G 24K 2K 2 (Ko-t2ko) 2(yxot oko) 2oKs ZoKy =the: ‘ E Reis ¢ = = ot, _ 1 (= =) .] dE dE, ... dE, (20)? 81X02, (2,4)! (ae)! 3! \W2s4e, —to confine ourselves at first to a first correction. Change the variable & by substituting for it &/+ Rox,/(«,+.*,), and integrate with respect to & between extreme limits—for &’, as for &, +o and —x. There results an expression cleared of & (as well as &), in which it will be observed that &,, &,... &,, 2’, enter the exponent only in (powers and products of) the second dimension; and enter the parenthesis on the right of the ex- ponential only in the first and third dimensions. Also the (imverse) coefficient on the left of the exponential, Q, which consisted originally of the continued product V/2 Kot, NW 2QokyT .+ MV Qo 7 is now transformed by substituting for the first two factors V2 (K+ 2%) 7 (one Vr being ejected). Rearranging the index properly we may similarly get rid of &,, the parenthesis on the right still involving only terms of the first and third dimensions of the variables, while in the coefficient on the left instead of the first three original factors we have now J2 (aK + ok) + 3K) T * Above, p. 116. coefficient. + The number of the element to which any « coefficient + EIN 2.x) being of the order unity, as we may say ; belongs is designated by a subscript on the left; the place small compared with ,/m. on the right being already devoted to the order of the § See note on p. 117. 16—2 124 Pror, EDGEWORTH, THE LAW OF ERROR. another /7 having been ejected). Continuing the process we reach ultimately the form 8 J 8 } y 1 —a'2/2k 7 A! av), ons [1+ Ba’ + Byx”) We may be sure, in the present case for which the regular law of error has been otherwise demonstrated, that the constants B, and B, are identical with those pertaining to that law. Next, let there be retained in the expansion of w terms of the order 1/./m involving second powers (and products) of &, &.... We have then &=-6-&...-E,+0 -Bn(—&—&—... +2’? —28,.(—-&—-& —...4a)&—... (with the degree of accuracy required for a second approximation) from which the expression dé, for da’ °82 be derived. Substituting these values in the expression to be integrated we az obtain an expression similar to that of the last paragraph; except that the exponent now contains, besides terms of the second dimension in &, &... &,, #, terms of the third dimension: such as — 28,27&,/2,«, or + 2°28,.€€,/2,«). But as each of these terms is affected with a small coefficient, the variables also being (in effect) confined to a limit of the order 1/s/m, we are entitled to bring down these adventitious terms from the exponent to the external parenthesis; thus obtaining for the expression to be integrated a form similar to that which was dealt with in the last paragraph; the form not being impaired by the factor d A TE (= 1 — 280! ~ 2Buks — 2Buk; — -..) Proceeding therefore, as before, we obtain a final result of the form 1 ; = pa w'2iak i n/3 ome” {1 + Biw'+ Box’). But we have not now the same @ priori assurance that this form is identical with the regular law of error. Indeed we may be sure that it is not identical with the law of error in the form which we have mostly employed—referred to the centre of gravity. For the mean first power is not zero, but By&" + Bo&% +... a quantity of the order 1//m, say PB, (+ quantities of lower orders). Put a’ =a’ + 8, and, expanding in powers of 8 and neglecting small quantities, we shall obtain for the locus of #” an expression of the form 1 —_ 9 &""/2k 17 ‘A a’’3], V2mrk he eee he Yy —d This expression must satisfy the conditions that (1) the area i ; yda (x, as we may say)=1, (2) | aw" yda (= a’) = 1, (3) | ay da (= wv’ 0)) = k; (4) I ayda (= a’ ®) j= a8) — 32’) 8 as 3a! a) 82 ue B, Pror. EDGEWORTH, THE LAW OF ERROR. 125 (quantities of the order 1/m (8?) being neglected). Equations (1) and (3) are already satisfied. We thus obtain, on performing the integration, two simple equations for B,’ and B,, affording for the solution a unique system of values. But the same equations are satisfied by the corresponding coefficients in the form ie ee 1 di = = 2k ey LE (3) i Vink E Bile a Therefore the compared loci are identical as to their constants, as well as in form. We have so far supposed that the a coefficients in the first term of the expansion of w are each=unity. But it is easy to pass from the simple case which has been treated to that in which the a coefficient of &, is a, by observing that wherever ,«, occurs in the preceding paragraphs a,7,«, should now be written; and (more generally) wherever yk, occurs a,'*?.«, should now be written. By parity of reasoning we may obtain the frequency-locus of « when account is taken of those terms in the expansion of 2 which are affected with the y's. Some difficulty in identifying the locus thus found with the regular law of error may be caused by the incident that the denominator in the exponent, viz—as in the preceding paragraphs—2k, is not now (as might perhaps be expected) identical with the mean square of deviation for z’. That mean square Sa Ee + 2% (Bym€." &." =i 3yiak E,” + 3ysnés E,” +502) (up to the required degree of approximation)=/ + (say)y. In order to compare the locus which has just been found with the law of error in the form that has been given, it is proper to substitute in the former, for k, k’—-y (where k’ is the mean square of error for 2’, and also—quantities below the order 1/m being neglected—for #”). Expanding in ascending powers of y we shall not introduce any new powers of #’*, in addition to the first and third which come in by the first correction, and the second, fourth, and sixth powers which the second correction introduces. As before it may be proved that the frequency- locus of x” must be identical as to its constants as well as in its form with the third approximation of the regular law of error. By parity of reasoning we may proceed to another and another approximation, observing that the coefficient of any power of «” outside the exponential, e.g. x, is now to be treated as made up of parts of different orders, eg. Bw +D,x, B, being of the order 1/)/m, D, of the order 1/m' +. When we go on to the case of several dimensions the question arises, how are we to understand an expansible frequency-function of several variables pertaining to a compound of numerous elements? The proper conception appears to be that the locus pertaining to each element, e.g. the rth, is of the form €=¢,(& 7, ...), assigning the frequency with * As may be seen by observing that Gif Yo," powers, Of a: dx * dz? + The small quantities 8 and y defined in the last where y) is the normal law of error (Part I. Sect. 2), and paragraphs will be made up of different orders. so the expansion in powers of y will involve only even 126 Pror. EDGEWORTH, THE LAW OF ERROR. which particular values of & and 7 are concurrently enjoyed by the rth element. The values of £7, .... are assumed by the rth element independently of ‘all other elements, but not in general independently of each other, the variables for each element being in general correlated. There is sought a locus of the form Z— ayes) assigning the frequency with which particular values are concurrently enjoyed by the compound magnitudes 2’ and y’..., each a (different) function of the elements, say, v=, Bacon 5 Mis May +-+5 enh TS Fe (Ba En eco8 “UP-Ghicook. ban) h these functions admitting of being expanded in ascending powers of the &s and 7's. Pro- ceeding by analogy we obtain, by multiplying together the ¢’s, an expression for the frequency with which m particular systems of values will be concurrently enjoyed by the m elements. Substituting for & and 7, their values in terms of the other elements and az’, y',..., attending also to the determinant proper to the change of ‘variables for &, m to x, y’*,..., we obtain as before an expression, with terms of the second dimension in the exponent and a descending series of groups in the external parenthesis. The exponent being put in the proper form, & and », may be got rid of by integration, and finally a result obtained which may be shown to be identical with the regular law of error both as to form and constants. Slight relaxation of the fourth condition. The modification of the fifth attribute which has just been obtained suggests an analogous modification of the fourth condition, that the variations of the elements contributing to a compound observation should be independentf. Consider the exponent in the expression for ,w in the preceding paragraphs, when first it was disturbed by our having put for &, not as in the simplest case{ a linear function of &., &,...a’, but one involving ascending powers of those variables. Just the same effect will be produced if we suppose that ,w—or rather what it becomes when the 2” therein is replaced by &+& +&+...+ &,—now denotes the law of interdependence between the values of the elements, and the problem is to find the locus of # the sum of &, &,... &, elements no longer perfectly independent, but correlated in such wise that ,w represents the probability that a particular system of values is enjoyed by the elements. As the compound need hot be a perfectly linear function of the elements, so the elements need not be perfectly inde- pendent of each other. The two kinds of imperfection may coexist. But the divergence from the typical ideal must in both cases be small. The suggestion that there is a general form of frequency-curve for a compound of elements interdependent in any way is met by * Above, p. 122. other groups. For then the sum of the elements in each + It need hardly be said that the fourth condition is group may be regarded as a single element obeying a not infringed when the m elements are interdependent in definite frequency-law; so that, n being sufficiently large, such wise that the set can be broken up intoalarge number the law of error will be fulfilled without further relaxation (n) of groups, the members of each group being correlated _ of the typical conditions. inter se but independent with respect to the members of all + Above, p. 124. Pror. EDGEWORTH, THE LAW OF ERROR. 127 the same reasoning as that which has been applied to the supposition that the compound is any function of independent elements. We saw* that if we are free to take a as any function of the elements, we can enable the frequency-locus to take any assigned form. Suppose that the relation of 2’ to the elements has been properly determined for this d da: purpose; and that (according to the last method) the values of & and —* having been thence derived, we obtain ,w in the form x (2", ioe eS ately Em); fen | | -sswdbidés ... dfn -x results in a function of 2 which is of the assigned form, say f(a’). Now substitute in y, for a’, &+& +...+&,; and let y (thus modified) represent the law of interdependence between the elements. If there is sought the frequency-locus of « the sum of such elements, it is found to be f(z)! Considerable relaxation of the third condition. The divergence from the type which has been discussed in the last section is to be distinguished from the interdependence between values assumed by the same elements in different observations. Such interdependence, if complete, would of course be fatal to the fulfilment of the law of error. If each of the elements had exactly the same value for all the observations there would be only one obser- vation! But a less complete degree of such interdependence is not fatal. Other things being favourable, there will still be fulfilled the approximate equality between the mean powers of the actual and those of the representative locus. Only the long run by which the mean powers of the actual locus are to be ascertained will now be longer. The Pearsonian criterion applied incautiously to such a system would yield a too unfavourable answer to the enquiry whether a given set of observations should be regarded as having emanated from— being a sample of—a distribution which in the long run would present the law of error. For the number, n’, which forms one of the data for that test is not now the number (+1) of the observations in a proposed set+; as, may be seen in the simple case where the observations are supposed to enter in pairs, the value of any element entering into the first, or any odd observation being the same as the value of that element which enters into the second, or generally the next, observation (,£;=,&; ,&)i:=,& 42). The observations forming each pair being thus identical we should regard 4m as the effective number of independent observations. The interdependence between the observations is more serious when they are con- sidered, not as in this paper, with reference to the law of frequency according to which they are grouped, but as constituents of an average affording the measurement of an object under observation. In this case each observation—divided by the total number of observations—stands in the relation of what is here called an “element” to the average. Accordingly a considerable interdependence between these observations will (by the last section) impair the law of error, and the rules which are based on it. * Above, p. 121. + Compare Mr Bowley’s remarks on the application of the criterion to statistics of wages, Journ. Stat. Soc. 1902, p. 338. 128 Pror. EDGEWORTH, THE LAW OF ERROR. Relaxation of the second condition. So far it has been taken for granted that throughout the long run formed by a series of observations the frequency-loci of the elements remain unaltered. But the conclusions which have been reached would not be affected by the relaxation of this condition, provided that the functions of the mean powers which have been utilised, k,, k,, ky, ... k;, remain undisturbed, however the individual elements change their form and « coefficients. For example, the first approximation for the frequency of an aggregate made up of m, elements, each formed by the number of pips on a domino taken at random, and m, elements, each formed by the number of pips turned up by a die thrown at random—the constant /; in the normal function which forms the first approximation to this law of frequency would be unaffected if in the course of a series of observations there were substituted for 198 dies, or any multiple thereof, the same multiple of 35 dominoes*. The average, indeed, of the total number of pips, presented by the m,+m, elements, would be altered by the change; but even this disturbance would be avoided if there were substituted, for the 198 elements formed by the casting of dies, 35 elements, each consisting of the number of pips on a domino plus the constant 1°8+. Lawity of the first condition. The condition that the varying values of the elements should be assumed at random limits the domain of our law less than might be supposed ; first, because the condition is fulfilled far beyond the sphere of aleatory phenomena from which examples of randomness are usually selected, and secondly because the condition need not be fulfilled perfectly. (1) Randomness is here understood as an objective} property of things, implying, inter alia, (a) that there is a “genus” or “series”§ of instances such that the proportion of instances possessing a certain attribute—belonging to a certain species—tends, as the number of instances is increased, to approach a fixed limit; (8) that a like series may be formed by taking as the genus each successive pair (or triplet, quartette...) of the original instances, the species being defined by the characters of the two (or more) constituent instances; (y) that like propositions hold good however a set of instances is arranged so as to form new series. As these properties do not depend on our “ignorance ” or feeling of uncertainty, so they are presented by phenomena which admit of the most exact knowledge and power of prediction. Such is the sequence of decimals in mathematical constants; the random character of which, in the case of 7, has been shown by Dr Venn. There are doubtless in the region of physics an infinite number of examples like the following :—The dates in minutes of high-water in the morning at London Bridge, given in the Nautical Almanack, have been taken for January Ist, April Ist, and July Ist, 1903; and according as the number of minutes in the date fell in the first, second, third, or fourth quarter of the hour, the number 1, 2, 3, or 4 has been put down. The * The mean square of deviation is for the domino (the + Cp. Venn, Logic of Chance; and Leslie Ellis, in sum of two digits) 16-5, for the die 2-916. Camb. Phil. Trans., yol. 1x. p. 605, on the ‘realistic ” + The average number of pips turned up by 198 dies nature of Probabilities. =34x198=693; the average number of pips on 35 § loc. cit. dominoes=2 x 4°5 x 35=315; 693 —315=378; 378/35=1'8. Pror. EDGEWORTH, THE LAW OF ERROR. 129 three numbers thus obtained have been added together. A like triplet has been obtained from January 2nd, April 2nd, and July 2nd; and so on (cases in which no morning tide is given for one of the dates belonging to the triplet being omitted), Other triplets are formed by similarly combining the dates of February, May, August, 1903. There are thus obtained fifty figures of which the ideal distribution is deduced from the respective probabilities that a single triplet should have the values 3, 4, 5... 11, 12 on the sup- position that high tide is as likely to occur at one part of an hour as another. The actual being compared with the ideal distribution by the Pearsonian criterion, the probability of random origin is found to be considerable, about ‘7. A not very different result is obtained from the like treatment of afternoon tides. (2) The illustration may be varied so as to illustrate imperfect randomness. If the triplets are selected—not from days separated by three months, but—from adjacent days, the periodicity of the tides impairs the random character of the result. But there are many methods of selecting triplets (or more composite elements)—from days that are neither very distant, not yet quite adjacent—of which it may be affirmed that they will probably result in an approximation to the ideal distribution. The character of periodicity—often consistent with, but sometimes fatal to, randomness—is illustrated by the digits of a recurring decimal with a long period. The writer has formed 180 decades out of the first 1800 figures in the period of the decimal which represents the fraction ;,,. These figures are given in the Messenger of Mathematics for 1864 (Vol. u. p. 1) in eighteen rectangular blocks, each block having ten rows and ten columns. For the first half of the series, the sum of the ten figures forming each column of a block has been taken; and for the second* half of the series the sum of the ten figures forming each row. If the figures dealt with obeyed a random law of frequency, the probability of any assigned digit occurring being ;5, it is to be expected that sums of ten digits would range according to an approximate law of error for which the mean square of deviation is 2A, (05? + 15°94 25°+35°+45*)=825. To test this the observed sums have been tabulated in a form convenient for the purpose of verification. Im the annexed table the first row presents limits demarcating the total possible range for a sum of ten digits, between 0 and 100, into seven compartments. The second row shows the number of observed decades in each compartment. The third row gives the corresponding calculated number, that is the mean of the numbers that would be given in the long run by batches of ideally random digits. The fourth, fifth, and sixth rows give the data for Professor Karl Pearson’s criterion of the probability that the observed set of statistics forms a random deviation from the theoretical frequency distribution assigned by the normal law of error. Employing the test we obtain for P, the probability that the given set is a sample of observations obeying the law of error, about 0-9. It is interesting to compare this result with one obtained from a set of 180 decades in other respects similar to the above, except that the digits were taken “at random” in the ordinary usage of the term—from mathematical and statistical tablest. The probability of this set having * The better to secure that the random distribution of half of the period. the digits should not be vitiated by the complementary + Part of the figures referred to in the Journal of the character (loc. cit., p. 39) of the digits forming the latter Statistical Society, 1885, Jubilee volume, p. 186. Wir, JOX, Ieee Ik 17 130 Pror. EDGEWORTH, THE LAW OF ERROR. Table showing the distribution of decades formed by adding successive digits in the periods of the decimal which represents +257. Limits of | | compartments |0 35:5 40:5 43°5 46:5 49-5 Ab 100 Ob d sip 26 28 99 23 22 36 23 decades Caleulated number of 26°604 29-223 22°367 23°611 | 22°367 | 29°223 26-604 decades (m) e 0-604 1-223 0-367 0-611 0°367 6-777 3°604 Cs 0°365 1:44 | 0-135 0°37 0-135 45:928 | 12:98 e2/m 0-01 0:05 0-006 0-015 | 0:006 1:57 0-48 x7 = Se?/m = 2:1. iP= IY). resulted from a random distribution proves to be about 02. The figures of a recurring decimal—the subject of exactest science—present the appearance of randomness in a higher degree than digits taken really at random! But if we vary the experiment by taking the first five digits of the decimal together with the five which occupy the 931st, 932nd, ... 935th places to form the first decade, and likewise for the second decade the five digits in places 6—10 together with the five in places 936—940, and so on, then the law of error entirely breaks down: there will be obtained a series of 186 compound observations, every one of which =9*! Yet though the condition of randomness above designated (y)t is thus imperfectly fulfilled, there are a great many ways of parcelling these digits which will probably result in a more or less perfect fulfilment of the law. The laxity of the conditions which give rise to the law of error may explain its prevalence over an immense range of phenomena—from the movements of molecules to the actions of men. * For a reason ably stated in the paper referred to (p. 39 et seq.). t+ p. 128. F. Y. EDGEWORTH. APPENDIX. PAGE Note 1. Orders of magnitude ; ; : : : : : : : : . 131 » 2. Degrees of approximation : : : t i : : ; : peels? » 3. Verification of the first method . : ; 5 é i é j . 133 , 4. Variant of the second method z F ‘ ‘ 5; - : 2 a “IBS » 5. Test of the third method : : Z : 5 5 : és : . 186 » 6. Variants of the fourth method : ‘ ; 5 < é 6 3 . 138 7. Substitute for the fifth method 3 : ; ‘ 5 : d \ a 13D. 1. Orders of magnitude. The reasoning about errors requires that several orders of magnitude should be distinguished. There is (1) the limit within which the “typical” (above, p. 38) element is in effect (as shown at p. 115) confined, the practical range, as it may be called, of the element, usually a small multiple of the mean deviation in 1 absolute quantity as given by the formula |£|=(S|£/’. (2) Corresponding to this limit of the element is the practical range of the compound quantity, a small multiple of its 1 standard deviation /k (or of its mean deviation |«|=(S)«|*)*). (3) It is evident, the number of the elements being large, that the last named limit belongs to a different order from the limit to which the compound observation might possibly extend, if every element contributed to the observation a value at the extremity of the element’s practical range, on one and the same side of the centre of gravity. (4) For the element too as well as for the compound there may be a possible as well as a practical range; as is evident in the case when the element is itself composite. (5) The admission of order (4) introduces a higher order of infinity to be enjoyed by the compound observation on the immensely improbable supposition that every element contributes to the observation a value at the (same) extremity of its possible range. (6) There is lastly to be considered the magnitude A€, or Ax (above, pp. 39, 113); which cannot be larger than the range of the least dispersed element—for example is equal to the range in the case of identical binomial elements—and may be indefinitely small. It appears to the writer that there is a propriety in treating this difference, however small, as still finite. Accordingly not much seems to be gained when, in certain versions of the Laplace-Poisson method, instead of the obvious quaesitum, what is the probability that a certain value of # meaning a certain multiple of Aw (+a constant) should occur, there is substituted the complicating enquiry, what is the probability that « should have a value between limits / + 7. 1/—2 132 Pror. EDGEWORTH, THE LAW OF ERROR. 2. Degrees of approwimation. How far the series presented by the law of error are available for purposes of approximation is an investigation which bifurcates according as the representative is compared with the actual locus, (a) with respect to mean powers, as above proposed (Part I. § 1), or (b) with respect to ordinates, as usually. In both cases the general rules as to the convergence and remainder of series are to be applied to the data. It will be found that in general ceteris paribus the series are less available, (a) the higher the mean powers compared, (b) the larger the abscissa of the ordinates compared. A consideration of the first term in the expansion of the law of error supplies the following tests, in the case of symmetry. (a) 2p, the highest power used for the purpose of comparing the actual and the representative locus, must be such that in the series at ee a (where k, = ky/(2k)), the first term within the brackets shall be a small fraction. Whence, if k, is of the order ~ in the typical case of irregular elements for which the « coefficients do not vanish, p? ought to be small with respect to m. (b) # the abscissa of the outmost ordinate that 4 is used for the purpose of comparison must be such that in the series y,+/h, }! — (where y, is the normal error-function with k& as constant) the second term shall be small with respect to the first. Whence k,(4—2x?+3x‘)—where x =«/V2k—ought to be small with respect to unity. When the loci are not symmetrical the conditions become more stringent. Example. Let the locus of all the elements be the same, namely the simple one first considered by Laplace, the horizontal line extending from —a to +4, with equation = 5 (AE being indefinitely small); and let the number of the elements be three*. We have then «=1@; m=— a‘; «,=1%a5;... and accordingly, for the aggregate of three 0 3 63 ) 5 2ai- a elements k,=1e; k,=—2a'; k,=4%a°;.... Whence for the representative locus we have +e = (5-2 sat3an)t ) = ne 2\2 ~ 2?" 3 4at iy Es If (a) we compare the mean powers of the actual and the representative locus; the second and fourth powers are, by construction, the same for both. The sixth power of the actual locus, viz. a° 94%, differs from the value given by the representative locus, not continued beyond the first term of expansion, by a°3%, a considerable percentage. If we proceed to another power, the approximation breaks down, the first two terms of the series being 105a%, —84a%. (b) The formula for the comparison of ordinates breaks down e7 2/202 ( ie Ol bo ‘ Dali 2 : : when—if not before— = oe (5= 2x? + 5") ceases to be small with respect to unity. * The example is considered by the present writer in the normal is sufficient to afford some approximation to the Journal of the Statistical Society (1900, p. 73) as showing law of error. A larger number is required for a good that a very small number of elements which are far from approximation. Pror. EDGEWORTH, THE LAW OF ERROR. 133 These propositions may be illustrated by considering the form of the actual locus: for . et il : which the equation is* y= — (3a?— a?) for values of the abscissa between +a; and beyond 825 : mde 1 = 3 those limits, Y = TGqa Oe F Caw + x"), the upper or the lower sign to be taken according as the positive or negative branch of the locus is designated. From these equations the mean sixth, eighth... powers can be found by integration. Also with the aid of proper tables+ the ordinates for the actual and the representative locus can be compared at several points as in the annexed table. Table showing ordinates (as multiples of «) corresponding to certain abscissae. Length of abscissa | 0 ple 1 Pp (in units of a) 2 \ First Approximation | 0399 | 03105 | 0-2418 | 01466 Next Approximation | 0°379 | 0:3092¢ Actual value | 0:375 | 0:3125 | 0:25 01572 3. Verification of the first method. The fundamental formula expressing the mean of 2", where w is the sum of numerous typical elements, this formula, proved above (Part I. § 1) by Generating Functions, may be verified by the Multinomial Law. First this law must be extended to Mean powers by the principle (employed above, p. 41), that if a, b, ¢ are independently fluctuating statistical quantities, the Mean of (axbxc...)=Mean of a x Mean of 6 x Mean of cx.... Therefore, if t! e=E tht... thm, oO =H SAVE £0, +5 SE MEM... EM, EM a, +... Consider first the case in which the &s resemble typical elements in that any mean power, eg. the ¢th, is approximately the same for one element as for another, but differ from the typical &s in that negative values are not assigned to these items; they are such quantities as the deviations on one side only of the centre of gravity contemplated at p- 116, which might be designated |&|. Then 2® is broken up into groups defined by the number of combinations between |£|’s. Thus in the first group the number of com- binations is that is smaller in the ratio t . m—t+1’ : m! in the second group r= mt F he UCU t!m—t’ WS and so on. The groups will form a descending series, provided that m is * Journ. Stat. Soc., loc. cit. Royal Society for 1900 (vol. xxxrx.). + For the calculations on which the first two rows of + The following approximation brings the value back the table are based there have been employed the tables again to the first ; showing that the series is not available given by Dr Burgess in the Transactions of the Edinburgh further. 134 Pror. EDGEWORTH, THE LAW OF ERROR. sufficiently large* (account being taken of the number of types, eg. two in the third group of the series, and also of their coefficients). An important case of this first theorem is where, for each of the positive quantities |&,|, we put the square of a typical element, viz. &,. We have thus, putting & for the mean of £2+£&2+...+&n2, that is the sum of the means £% + &®+...+ En, ht = (E, + &® + 2. + Em) = (EP + EP +... + Em?) =f! DEF .. .&E,2) 45 a 1, DEE, @ £4 (E03)? + Whence D£,&" ... &° == kt, to a first approximation. Also a pA” 1 EP, 4 (EP = es DEE... E°, . x &(E,”)? approximately, wo =(by the former approximation) t(t— 1), kt (&,)? approximately. ill = (2) \2 i z Therefore é&,£,°)... &,°) So ae 2 ae) , to a second approximation. Next, considering typical elements, since &”, &" ... &m” all vanish, : &° (*) E (2) ~ &,°) &,") &0) a Eu) 2?) = pS a Se 4. SE + pl SS ow Og @ £2 (2) (3) (3) yghouees £93 CO he pla op ot an el sata eielnle Substituting for the continued products of mean-squares the approximate values given in the last paragraph, we have 2?) = 2p! kP oe 2p! jp melts) 2p! ke DE, Qp! kP-s LE, mE, 8) pla? p—2!2P22 2 'p—2!9P2 41 ‘p—B3l2e 318! Qn! kP Dy! ke Qp\ kes 1 (2E,% 38 4p 4 @) (Ce) : OD veer p= x(a — 3(&,0))?) + — =a 97-85 s 31 (approximately). Replacing Y&,° by k(=X«,) and =(E,“ —3(E,%)) by (= X«.) we obtain the result already deduced (Part I. p. 43). The expansion of odd powers may be similarly verified. The principle may also be extended to the case of two (or more) dimensions; verifying the expressions which have been found for (XY, Y)®) (above, p. 118). Some difficulty may be caused here by the incident that al/ the terms involving mean first powers do not now vanish; there survive, along with terms involving only second powers, terms of the type (2) (2) = (2) (2) (2) 2p! Po a Er = a) Sa x 2p’ H, Fy, I TOr To VCR Mah gous TO 2) 4 i ap oh Aw Hw Per * It must be large enough to cover any irregularities in the gradation which may be due to the absence of perfect equality between the mean powers of the different elements. Pror. EDGEWORTH, THE LAW OF ERROR. 135 (supposing e.g. ¢ and ¢’ even) where the first powers do not vanish, as they form factors (Bp, Hy), (Spi, Hpi)” (Gn the notations of p. 116), say A, and A,,,, which, the coordinates for the same element being in general correlated, do not vanish. As there may occur any even number of these A’s as factors of constituent terms in the pre- dominant group of the multinomial expansion with which we are now concerned, it may seem at first sight that the character before ascribed to that group—a sum of products of mean squares—is no longer maintained. But it must be remembered that, as the axes X and Y are (with respect to the aggregate) principal axes, (X, Y)"", that is a aac Demers ), vanishes. Accordingly ZA,A, has not the weight to which it seems m(m— 1) entitled on account of the number of its terms. It has terms, any one of 2 which may in absolute quantity be of the order =*H*; yet the algebraic sum of these terms is equal to half the sum of only m squares, viz. Ay?+ A,?+...+ A)? (since (ZA)? = 0). If then we substitute in the last written expression for a portion of (X, VY)“, when t— 2p — 2p. (2) Ul 39) ss Ss Pa x S2y’! H,° H,," Ay. 22p! Sr er ot a ey x SA,Ag, we see that this portion of the expansion does not rank with one which is made up of square factors only. The former is of the order 1/m as compared with the latter. Like reasoning is applicable to products of more than two A’s. Thus 2A,A,A;... Aog does not contribute to the terms of which it is a factor, a weight proportioned to the m! Bale but only to the number of terms in the predominant group of (2A)? that is : 7m ° r=m q 5 : . : sum of the other terms in (= A,) which, m being very large with respect to gq, is of an r=1 4 order ~ compared with the number of terms in the seemingly predominant group of that series. So the A’s do not affect the predominant group in the expansion of (X, VY) with which alone the first approximation is concerned; they are taken account of in the third approximation by the coefficients of the form ,,K, and in subsequent approxi- mations. 4. Variant of the second method. The general solution of the leading partial differential equation in Professor Crofton’s method, viz. dy _ dy 1 dk 2 dx?’ Pc may be written ae V it 9 Qs where ¢@(k) is an arbitrary function involving the symbol a Ls and Q denotes the significant part of the expression, the part not affected with that symbol. In the case before us we know by Professor Crofton’s subsidiary condition * Cp. Forsyth’s Differential Equations, Art. 256. 136 Pror. EDGEWORTH, THE LAW OF ERROR. that $(k) is of the form pth) and by theorems above given that y involves the coeflicients k,, kh)... in such wise that (az) ¥= (-si(ae)) 9 (i) ¥= Ge) ) 9 These conditions will be satisfied by putting for ¢(k) aorta) ebm (ea) 1 V2k° where the condition that, when h,, &, ... vanish the integral of y between extreme limits should be 1, gives A=1/)/7*. It will be observed that the operator (in ax) constituting #(k) is the same operator used above in Part I. to express the law of error, say F (): dx when for e is substituted vi ae We have thus a variant form of the law of error was 2 au etl anv a F(s/25) we. This result may be verified by changing the order of the two operators on ; NV 2Qark Misa ae ee Rei! ; substituting in F, thus transposed, for 2 ag (as we are entitled to do by the ae dy ide : : eae: 34, equation Fie 5 aa) and moving Q, as after this substitution is legitimate, so as to follow F. We have then d = 32 1 = AGEN I a &) ae Va Jaan el aes the original form. 5. Test of the third method. The azopiac of Leslie Ellis and Professor Czuber respecting Laplace’s method+ are not damaging to the version of it above given. They take as a test-case that in which the elements have each the locus, in our notation, n= i erly, 2y where the upper sign is to be used for & positive, the lower for & negative. The parameter y does not figure in the form given by the eminent authors; they treat as unity a magnitude ae 1 x + Articles on ‘‘ Method of Least Squares,” Camb. Phil. * Since ae a/' BE ae ok. Trans., vol. 1x. p. 605; Theorie der Beobachtungsfehler, v is pp- 267 et seq. Pror. EDGEWORTH, THE LAW OF ERROR. 137 which has been above treated as of the order 1/,/m (the standard deviation of the composite locus being treated as unity). Professor Czuber investigating, on the lines of Laplace, an expression for the probability that the aggregate of elements will lie between certain limits, questions (loc. cit., p. 267) whether the results obtained by the approximations employed are not mere formal but actual approximations (nicht blosse Ndherungen der Form sondern auch Néherwngen dem Werte nach sind), unless some additional assumption is made (loc. cit., p. 271). This doubt appears to be justified; but it does not touch the version above offered, in which it is proposed to supplement the reasoning as usually presented by additional—in general justifiable —assumptions: namely, (1) that the elements are such and so numerous that the coefficients k,, ky,... are each small in relation to the corresponding powers of k, the mean square of deviation for the aggregate; k,, k,,... and likewise k,, k;,... forming series descending in order of magnitude (above, p. 42); (11) that the abscissae of the ordinates for which an approximation is sought should not be large with respect to /k (above, pp. 131—2); (111) as a consequence of (I), that the elements may be treated as virtually confined to a range which is small with respect to /& (above, p. 115). The third assumption removes any scruple which might be felt in expanding x(a) (above, Part I. § 3) in ascending powers of a and mean powers of an element. In the case before us let us employ the rule suggested above (p. 116), and take as the limit on either side to which the range of the element is practically confined, two or three times OP where OP =(\&|"):*. We have for the mean tenth power of deviations on either side 10! y"; and, for OP, y log_, 0°6559, which, multiplied by 3, > 13y. Let us be content with 10y as the limit on either side of the centre; thus leaving out of account a proportion of area less than a twenty-thousandth (!); the operation of which cannot sensibly affect our results. Attending to the orders of magnitude involved in y (a) and (y(a))” (loc. cit.) we cannot doubt the legitimacy of the step which gives he ie Fis 1 | er etatsiht- cos arda for the required function. By assumption (1) we are entitled to differentiate with regard to k, under the sign of integration, and expand in powers of k,, in a parenthesis outside the exponential; thus obtaining a series of definite integrals Th 1 y Wn GEO: —ak a sy —a2k = Sawn Pa PIL =| e cosarde + ihn] ate cos awd +... Yor qyha ga te: By assumption (11) such values of w are to be employed as (tested by the usual rales for series) make the above-written series convergent. Ellis first obtains a series in ascending powers of u (his equivalent for our «) and descending powers of xz (his equivalent for our m) representing the probability in the case proposed that the sum of n elements should be w (between w and w+du). Comparing this actual locus with the normal law of error he expresses a doubt about their coincidence, except upon the supposition that is (not merely large, but) infinite. He remarks (comparing the two Wor XOXe PAR ale 18 138 Pror. EDGEWORTH, THE LAW OF ERROR. : Uae yp) ‘ functions) “we retain op and neglect —_, although unless w be large, the former term is n of the same or a lower order of magnitude than the latter.” The explanation of this difficulty is that we are concerned only, in a first approximation, with the case in which u is of the order Vn [a of the order Vmy]. For wu of this order it is quite proper to “retain 2p 2(p—l) ‘ . z ae _ and neglect = = ” The next order of magnitude (in the case of even loci) is A 1 Y : : 1 ¢ 1 F our —, or ——]| smaller in the proportion —, that is, of the order ~~. When uw is m vm n Vn thus small, both the compared terms are to be neglected in the first approximation formed by the normal error-function. The actual and the representative function give the same value : 1 1 for w[=a]=0 when fractions of the order 5 |= | are neglected. If greater accuracy is required the next approximation must be taken in; furnishing the correction $k, x y% ; cnet ; 1 (where y, is the first approximation), for w zero or a quantity of the order ak 6. Variants of the fourth method. In applying the condition that the sought function must be reproductive (Part I. § 4) it was at first taken for granted, as a result of the funda- mental theory (cp. p. 46), that in a first approximation we are concerned with only one constant (k). Other constants (/,, k., ...) were introduced later. Otherwise, we might at once introduce one or more additional constants, say 7, and determine w (Joc. cit.) so that map (a, j) =" (pa oj): A general solution of this functional equation, found by an extension of Boole’s theorem referred to above (p. 55), is Aat+ Bart)" + Cj", where p+q=1. This general solution is required for the application of the fourth method to two dimensions. But for the present purpose, if we may presume from the fundamental theorem that the constant is 4, entering into the second approximation only in the first power, the general solution for yw reduces to Aat + Ba?th,. By the first approximation in the text we have ¢=2, p=m. Bringing down Be?*k, from the exponential we have for the first correction, if y is the first approximation*, dey y | * dat ? assuming that pt must be an integer. We may employ the condition that 4, enters the sought function over k,*+ to determine that pt=3; and accordingly o=1/\m. (Real part of +1, or V—1) Bk Again, the investigation of the first approximation in the text may be varied by taking as subsidiary to the condition of reproductivity the partial differential equation which was taken as primary in Prof. Crofton’s method, viz. dy _1d’y dh = 2 dae (above, p. 46, Part I. § 2). * See p. 53. + Above, p. 46 and ep. Phil. Mag. Feb. 1896, pp. 95, 96. Pror. EDGEWORTH, THE LAW OF ERROR. 139 Let the function, supposed to be reproductive, be i a aa) (5), where c=q Vk (above, p. 56). / We have then, beginning with the case of symmetrical elements, x (a) = | 5 * f (=) cos axda. af —o Put taken in the text we find that (ca) must be of the form a(ca)'; and accordingly =x; and it is seen that y(a) is of the form ¢(ca), say ev). By the first step Ais c im =| er(9Vka) cos axda. ZEHO _ dy 1d t f 4 : Now introduce the equation 3 ey and we have 5 ag'atk2 =—4a*; which can only be y s tt A satisfied by making t=2,aq?=—+4. (Substituting this value of ¢ in the expression for y, we see that a must be negative in order to secure a finite result.) Integrate with respect to a and we obtain the normal function, A (and ag?) being determined by the conditions that the mean zero power = 1 (and the mean second power =k). The case of asymmetry may be analogously treated by the use of et¥—* for the first cosax, and e-Y—* for the second cosax, in the preceding investigation (see the text, p. 56). Once more, the last written partial differential might be treated as the leading one, and the condition of reproductivity as subsidiary. One of the general solutions of the equation is of the form | da | e-watk cos a(a— r) (A) dr*, J0 —2 where y(A) is an arbitrary function. This may be regarded as the result of superposingt a statistical quantity having the frequency-locus (x) upon another quantity having the lee ater : : frequency-locus = | e-*2k cos axda, that is, the normal law of error. In order that this form /0 should be preserved when another form is superposed, that other must be also normal (one constant only being admitted as is proper in a first approximation, above, p. 138). Eg. if vv (a) were of the form | e°@* cos ax, where t is other than 2, it would not (though itself 0 of the reproductive family), when superposed on the normal curve, produce a reproductive function of the simple species proper to the first approximation. Therefore (zr), as well as the other component, is normal; and therefore the compound is normal. 7. Substitute for the fifth proof. The extension of the law of error to functions capable of being expanded in ascending powers of the elements was, in the text (above, p. 122), based * Forsyth, Differential Equations, Art. 258, t+ Above, p. 45. 18—2 140 Pror. EDGEWORTH, THE LAW OF ERROR. on a postulate which seems particularly appropriate to such a function; that the range of any element is (in effect) small. The proof may also be effected independently of that premiss, on the lines of the first method (Part I. § 1). Let «’ (as above, p. 123) be an expansible function of the elements; the first term of the expansion being, as we may put without loss of generality*, & + E+ woe t+ En (= a). Consider any subsequent term of the expansion, eg. wé&&£, (when mw is small), making abstraction of other terms after the first. The mean powers of 2’, each divided by the corresponding factorial, are generated by a function of which the expansion in ascending powers of @ is 1+ 02’) + = 62a?) + = Bz’) + ..., where eM) =o + pre, VEMEW =O; a?) = gl +e (2&,&&)” 1 =k JL PLE, EAE?) =k+ we 31° (terms which vanish being omitted). Likewise fey ie aw) = Bh? + ky + QaydEVESES +... = BK? + hy + Apk terms of an order lower than yp, as well as those which vanish, being omitted). And so on. BK 8 Consider, as the generating function of mean powers, val wes i 1 ; Exp. log E + 0? 21 G + 3 ! wie) a5 6 31 (ky) + (che rae se 4k?) + | “ The character of the law of error will be preserved if the coefficients of , 6%, @4,..., say ky, ko... continue, when referred to the corresponding powers of &,, to form a descending series. This may be verified by expanding the above-written logarithm in powers of @. It will be found that no power of @ after @ is affected with a coefficient of so high an order . : 1 : : . : as w if w is of the order mt The generating function up to the third degree of approxi- mation is 1 1 1 pil eee Pay O4 (Keg + 4k) " 3 : : ee ; aie as may be verified by comparing the value of Il x") as given by the coefficient of @? in the expansion of this function with the value obtained from the equation a?) = (e+ wLEEE)? = a) + 2p 2p—1! CLE, PEPE .0 vee Epo EVE E?, ai 2) 21 2! * Tf the first term is + No exception to this statement is made by a term of af, +a.é.+&e., put §&=§//a,, &=h'/ao...« the form @cuk*Dé,4) (c a small numerical constant) since (Cp. above, p. 125). DE, is of the order 1/m=u. Pror. EDGEWORTH, THE LAW OF ERROR. 141 where c is the number of different arrangements that may be made with the same (p+1) elements so as to form the same product of squares, viz. (p+1)p(p—1)/3!, = 2°) + 2p! pkPt/2P 3! p — 2}, as it ought. Analogously, verification may be obtained with respect to any other terms in the expansion of a’, e.g. ADEZ or vEE,E.E,E,; where X should be of the order 1/Wm, v of the order 1/m!; and it is to be observed that 2”, in the first instance, does not vanish, but = AA. Corresponding verifications are to be obtained in the cognate case of elements not perfectly independent* ; when in that expression for the mean value of e®** from which the generating function is derived (above, p. 41) the means of products such as &&, &&2, &&&,, &2&2, &. (in the notation above used (&, &)>, (&, &)%®, (&, &, &)%%%, (&, &)) differ by small quantities from what they would be if the elements were perfectly independent, viz. respectively, 0, 0, 0, & 0%. In both cases the propositions admit of generalization. * See p. 126 above on the analogy between curvilinear functions of independent elements and a linear function of interdependent elements. VI. Memoir on the Orthogonal and other Special Systems of Invariants. By Major P. A. MacMaunon, Sc.D., F.R.S., Hon. Mem. Camb. Phil. Soc. [Received 6 January 1905.] PANG ume le PROFESSOR ELLiorr has constructed a complete syzygetic theory of the absolute orthogonal concomitants of binary quantics; his method is not symbolic, but it is quite convenient for obtaining the fundamental covariants of a quantic whose order is small or of a system of such quantics. In general the process leads to a Diophantine Equation which is difficult to handle for an order which is not small. I propose, in the present research, to obtain results of a general character by the employment of a symbolical calculus involving imaginary umbre. I obtain an inferior limit to the maximum degree of an irreducible covariant of given order, appertaining to a quantic of given order, and in certain cases I shew that there are also superior limits, but I have not succeeded in establishing this for the general case. For the first three degrees I obtain the actual numbers of irreducible absolute concomitants of the quantic of general degrees, and also some results in respect of the fourth degree. These results enable me to obtain the fundamental syzygies of the third degree. Gaal Let the binary quantic be C—O Cnr and the transformed quantic, due to the orthogonal substitution, Ay” = By” = Cy” FS eeey then it is easy to shew that Dye Uae, — en (CXG ee wae) a, + ta, = e* (A, + 7A,). I write 2 +1%,=F,, %— ta, = &, and I take new complex umbre, linear combinations of the old umbre, viz.: I write +d =%, Gy —1dy= OM, b, + ib. = By, b, — tbs = Bo. Masor MACMAHON, ON SYSTEMS OF INVARIANTS. 143 The general expression of a non-absolute orthogonal concomitant of degree @ and order e€ is as Ct Bi [spe war fates EP, where there are @ different symbols a, £,.... For the form to denote an absolute concomitant the second of the exponents of symbols with suffix unity must be equal to the sum of exponents of symbols with suffix two. Hence S +54... 59+ €—p=On—s,—S....—Set p, or Xs=4(0n—e)+p. Since &,€=a22+ 22, is a covariant, and we are concerned with irreducible forms, we may put p=e and take the form to be a,* ah Bi BS ss ie. where =s=4(On+.e). The weight w may be taken to be 4(@n —e), so that On — € = 2w. The form BERTI SEY SEE pn FEO for different values of s,, s.,..., satisfying the condition Xs=4(On+e), denotes the asyzygetic invariantive forms of the degree @ and order e«. When e>0 we have the associated form a7 a5 Be BS Ey Es, obtained by interchange of suffixes; hence in this case the invariantive form connotes two concomitants. When e«=0 there is a one-to-one correspondence between the invariantive forms and the invariants. For example the linear quantic has the invariantive form of degree 2 a, a2" 8, Bo, which is unchanged by interchange of suffixes, and therefore yields one and not two in- variants; in fact a, 8. = (a, + ta2) (b, — tz) = a —7 (ab), wherein (ab) is a vanishing form. The covariantive form may be written P+iQ, 144 Masork MACMAHON, MEMOIR ON THE ORTHOGONAL AND and denotes the two covariants P,Q; the reducibility of P and Q depends upon the reducibility of P +7Q and conversely. The symbolic identities in the umbre, a, 6, c,... can be very simply obtained from the complex umbre a, B, y,..... Ex. gr. from the identity 4%. Bi B,=% B.. Ai, we derive Agdy = {ay — 1% (ab)} {ay +7 (ab)} = ay? + (ad)? ; and from MO. By Bo. V2 = %Be- Bie « 1%, we derive AabyCe = bela — My (bc) (ca) — b. (ca) (ab) — Cg (ab) (bc), abe (ca) + beCa (ab) + Cady (be) = (ab) (be) (ca), by equation of real and imaginary parts. It will be convenient at this point to have before us the irreducible invariantive forms for quantics of the first six orders for comparison with Elliott’s list and for easy verification of general results to be presently established. For brevity I will denote the form a Ge B® Joye ee Es, by (GE coe HBC) Linear Quantic. The irreducible forms are (10; 0), 1; 1), &&; where observe that the covariantive forms other than &é&, connote two covariants; in fact (1; 1)=a,& =a, —2 (az), and we have 1+2+1=4 invariants, viz. ay, Az» (ax), a? =F Xo". The table is Order 0 1 2 0 1 2 ed 2 = 2 1 Quadratic. The irreducible forms are OTHER SPECIAL SYSTEMS OF INVARIANTS. 145 yielding five forms which in the umbre a, 8, c, ... are Qa, O, G3, Az (wa), Ly. I give these in the most suitable forms, not restricting myself to those which are immediately obtained from the complex umbre. The table is Order 0 1 2 0 1 oO 2 01 1 2 =) Bh al | Cubic. The irreducible forms are (S0510)) (21550) (SUL 0) (222070); (2; 1), (811; 1), (1; 2), (3; 3), &&, viz. 4 invariants and 9 covariants. The table is Order 0 1 2 3 0 | 1 1 2 2, oOo o | ae) 2 (=) a 2 | YB ae ae | Quartie. The irreducible forms are (2; 0), (40; 0), (31; 0), (411; 0), (830; 0), (3; 2), (41; 2), (4; 4), &&.,, viz. 5 invariants and 7 covariants, Won, VOX, IPA IE 19 146 Mason MACMAHON, MEMOIR ON THE ORTHOGONAL AND The table is Order 0 1 2 3 0 1 euler 2 2 30 Ml aae 2 3 2 Quintic. The irreducible forms are (50; 0), (41; 0), (82; 0), (5311; 0), (4420; 0), (5221; 0), (4330; 0), (4 (552111; 0), (444300; 0), (522222; 0), (333330; 0), (55511111; 0), (44444000; 0), (3; 1), (521; 1), (440; 1), (422; 1), (55111; 1), (52222; 1), (51; 2), (42; 2), (6222; 2), &E, (4; 8), (522; 3), (52; 4), (5; 5), viz. 15 invariants and 27 covariants. The table is Order 0 1 2 3 4 0 1 1 ‘ 2 2 2 3 4 2 3 6 2 o Sy 6 2 o a 5 4 6 4 if 8| 2 (3331; 0), 0 tt an OTHER SPECIAL SYSTEMS OF INVARIANTS. 147 Seatic. The irreducible forms are (3; 0), (60; 0), (51; 0), (42; 0), (621; 0), (540; 0), (522; 0), (441; 0), (6222; 0), (4440; 0), (6411; 0), (5520; 0), (66111; 0), (55500; 0), (4; 2), (61; 2), (52; 2), (622; 2), (550; 2), &&, (5; 4), (62; 4), (6; 6), viz. 14 invariants and 17 covariants. The table is Order 0 1 2 3 4 a 6 0 | [es ae 1 il 2 | 2 2 | a2} 3 4 | | 2 | ee | | ae ro Baia 4 | 4, hd | | 5 2 sal | § 2. It has been shewn that the asyzygetic forms of given degree and order e for a quantic of order n are given by aFa." "8B"... Ef, where the number of symbols a, 8, ... is equal to the degree @, and S +5.+...+8,=4(On+.e). Hence the number of these asyzygetic forms is equal to the number of partitions of 4(6n + €) into 6, or fewer, parts, none of which exceed n; the number is, in fact, the coefficient of at ents) in the ascending expansion of : (I—a) (—az)... 1—aa")’ 19—2 148 Masor MACMAHON, MEMOIR ON THE ORTHOGONAL AND or, denoting 1—* by (s) for brevity, this is the coefficient of a} +? in (n+1)(n+ 2)... (n+ @) (@)i@2).... (6) The form (Tele Rim othe opm ea tan is reducible only if it be factorizable into other forms. This fact enables us to specify the conditions which lead to irreducibility and thus to specify those partitions of + (On + €), into @ or fewer parts, which give irreducible forms. If the form can be exhibited in two ways as the product of irreducible forms of lower degree and order, we have evidence of one or two syzygies according as e=or>0. These observations will be verified as the work proceeds. The first of them enables us to find forthwith an inferior limit to the maximum order of covariants of given degree to a quantic of given order. A preliminary observation is that, if the form a8 a8 B= [span ae Be a" ash Bt [spo Aer E91, of degree 6,(<@) and order e(0. Consider the form a a? [opie fore via yo Dias E,% ; there being n—e symbolic factors with exponent n—1. It is of degree n—e+1 and order 2e. Denote it by ann — le) Jor (2a — Ia") De): and suppose a factor of the form to be : (S)85.--= 895 2); where O609. Inferior Limit to the maximum degree of an irreducible covariant of given order e€ to the quantic of wneven order 2n+1, € being > 0. Consider the form qu a,° fey [Sher haat = E.¢ : there being 2n+1-—e symbolic factors with exponent x. This form is of degree 2n+2—e and order e; denote it by PASE gy hinme ark oye (Op se IL pete. ya) Suppose a factor of the form to be (Gisoosshs of degree @ and order ¢’, where 6<2n+2—-—c6 e 0. OTHER SPECIAL SYSTEMS OF INVARIANTS. 151 § 3. IRREDUCIBLE INVARIANTS AND COVARIANTS. Degree in coefficients 1; order of quantic 2n+1. The form of symbolic product is as ae aoa where s=nte+l1; leading to CHL EES There is thus one irreducible symbolic product yielding two irreducible covariants in respect of each uneven order. The order of the covariant evidently cannot exceed the order of the quantic. Degree in coefficients 1; order of quantic 2n. The form of symbolic product is asa ae 2 } a3 where sS=n+e; leading to ECE giving one irreducible invariant and two irreducible covariants in respect of each even order. The order of the covariant cannot exceed that of the quantic. Degree in coefficients 2; order of quantic 2n +1. The form of symbolic product is 810,20 3-81 BS Ben +18 £26 where 8, +5. =2n+1+e. The number of asyzygetic products is equal to the number of binary partitions of 2n+1+., zero being admissible as a part, and no part exceeding 2n+1. The product may contain a factor ante Hg noe £241 and then S=nt+e +1, ss =n+e—e. Hence, for irreducibility, it suffices to have no part of the binary partition equal to n+e +], for all values of e’ from 0 to e—1; that is, no part must be taken from the series n+1, n+2,...n+6 The binary partitions satisfymg these conditions are Qn+1,¢; 2n,e+1;...n+te+1,n; n—e+1 in number. : 152 Mason MACMAHON, MEMOIR ON THE ORTHOGONAL AND Hence there are nm—-et+ 1 irreducible symbolic products and for es, WP 8, con the number of irreducible covariants 1s n+1, 2n, 2(n—1), 2(n—2),... respectively. Generally for the order 2e, (e>0), there are 2(n—e+1) irreducible covariants. The theorem given above in regard to the inferior limit to the maximum order shews that, for degree 2, the order of an irreducible covariant is at least equal to n; hence the results are consistent and in this case the inferior limit is also the superior limit. Degree in coefficients 2; order of quantic 2n. The form of symbolic product is a1 a1°"—% 8,8 B82 Pe, where 3 +5 =2n+e. The number of asyzygetic products is equal to the number of binary partitions of 2n +, zero being admissible as a part, but no part exceeding 2n. The product may contain a factor ante ata fae and then S%=N+e, S=n+e—e. Hence, for irreducibility, it suffices to have no part of the binary partition equal to N+e, for all values of ¢ from 0 to ¢; that is, no part must be drawn from the series nm, n+1,...n+e. The binary partitions satisfying these conditions are 2n, ¢; 2n—1, e+1;...n+e4+1, n—-1; n—e in number. Hence there are n—eE irreducible symbolic products and for e=0) eS oe the number of irreducible covariants is n, 2(n—1), 2(n—2), 2(n—8), ... respectively. OTHER SPECIAL SYSTEMS OF INVARIANTS. 1538 Generally for the order 2e, (e>0), there are 2(n—e) irreducible covariants. No irreducible covariants exist when e>n. Moreover we have established above that 2 is an inferior limit to the maximum degree of a covariant of order 2n—2 to a quantic of order 2n, or, the same thing, that if the degree be 2 the order of an irreducible covariant is at least equal to 2n—2. Hence the results are consistent and the inferior limit to the maximum order of an irreducible covariant of degree 2 to a quantic of order 2n is also a superior limit. Degree in coefficients 3; order of quantic 2n +1. The form of symbolic product is SN 6a OFS Fac Say eta es eae where 8, + S + 8,= 3n+e42. We know that there is one irreducible product of this degree when ¢ is equal to n—1 and it will be shewn subsequently that ¢ cannot exceed n— 1. Put therefore N=v+e, so that 2v is the excess of the order of the quantic over the order of the covariant. The symbolic product is then aig? HEH 8 B2vt2H-Sery,85 nyu tet iss £2641, cere S$, +S.+ 8,= 3y + de + 2. We are concerned with the partitions of By + 4e+ 2 into 3 parts, zero parts being admissible but no part being greater than 2y + 2e+ 1. If the product be reducible there must be a factor asa, Hes 21, where S=v+tetetl and e in = Co,_, ls (1-2) l+ez (1-2? (1—2*) _ (sr _ j(@aere ll = Cos; OTHER SPECIAL SYSTEMS OF INVARIANTS. WSS) Hence the number of irreducible symbolic products is v+1\. ("3 )) and the number of irreducible covariants of degree 3 and order 2e+1 to the quantic of order 2n+1 is (| or (n—e)(n—e+4+1). As a verification it is not difficult to prove that the coefficient of asgsntet2 in the expansion of the fraction (1 — aa") (1 — aa") wee (Us () (—a)(1—ae)...(l—aa") is equal to }(n—e)(n—e+1), and this supplies us with a proof that the order of an irreducible covariant of degree 3 cannot exceed 2n — 2. Degree in coefficients 3; order of quantic 2n. The form of symbolic product is a0," By BA on HE where S +5 +5,=3n+e. For irreducibility the ternary partition of Bn +e must have no part drawn from the series n, n+1,...n+6 Putting n=vt+e, consider the ternary partitions of 3v + 4e no part >2v+2e and no part drawn from the series v+e, v+tetl,... v4 2e. The possible parts are in two series: first series ON 2s aa wie — lye second series yv+2%e+1, v+%e+2,... 2v+2e; and, clearly, we must take one part from the first series and two from the second or two from the first and one from the second. On the first alternative the parts may be y—s—t—2 from the first and v+2e+1++s, y+2e+1+¢# from the second; and on the second alternative, y+e¢—s—1, y+e—t—1 from the first and v+2e+s+t¢+2 from the second. 20—2 156 Mason MACMAHON, MEMOIR ON THE ORTHOGONAL AND In both cases a partition corresponds to every partition of v—2 and lower numbers into 2 or fewer parts. Hence the enumeration is given by 1 2 Co aa” in (1—2)(1—a)(1 —az) (1 —aa*) 1 1+2°+ 27 — 9 » = — = > S—— - Coys; daaFd-a) 2iCo=e (i aip If v be even this is l+z 2 Co, (v—2) d—ay =2 (#03) 42 (#7) 9 Vv. bole If v be uneven it is Qx (=a) ae =a c (v+ zu 2 = 2 Coy =4(v-1). Hence the number of irreducible symbolic products is $y* or $(v*—1) according as v is even or uneven, Thence the number of irreducible invariants of degree 3 to the quantic of order 2n is $n? or $(v?—1), according as m is even or uneven; and the number of irreducible covariants, of degree 3 and order 2e(e>0), is equal to (n— ey or (n—e??—1, according as m—e 1s even or uneven. This result is clearly consistent with the previously established theorem which states that for the degree 3 the inferior limit to the order of an irreducible covariant is n— 2. The facts that have been discovered in regard to the quantic of general order are complete as far as the third degree in the coefficients inclusive and we may now enumerate the syzygies of the second and third degrees in the coefficients. The syzygies, between symbolic products, are of two kinds, (i) those which involve the covariant ££; (11) those which are free from this covariant. To obtain a syzygy of the first kind it is merely necessary to multiply a covariantive symbolic product by its conjugate to obtain a form which is divisible by some power of &&. I propose to avoid these by restricting myself to symbolic products of the form a," CHC a Ee ; OTHER SPECIAL SYSTEMS OF INVARIANTS. USL and I proceed to the study of the syzygies of the second kind appertaining to such forms. The syzygies of the first kind might be made the basis of the study of the whole question. By the method of the present paper they are not necessary. The table of fundamental symbolic products is, for a quantic of order 2n +1, Order 0 1 2 3 4 5 6 7 8 9 0 1 | | eo 1 1 | 1 1 1 | = | A 2} n+1 n | n—1 n—2 n—3 | | n+1 7 n—1 n—2 m—3\ | : (3°) (3) (3) ("3°) ("3") | The corresponding table of invariants is obtained by multiplying all numbers except those in the first row and first column by two. Of any degree-order we have the known theorem which states that the number of syzygies is equal to the excess over the asyzygetic forms of the sum of the number of compounds and the number of irreducible forms. There are clearly no syzygies of degree 1. SYZYGIES OF DEGREE 2. Order of Quantic 2n+1. The number of asyzygetic symbolic products of type S02" 3-8 9,84), 2n HE 2 is equal to the number of binary partitions of 2n+e-+ 1, no part being greater than 2n +1. This number, when e¢ is of the form 2e, or 2e,+1, is equal to n—-@Qt+l. Moreover there is one irreducible symbolic product of degree one, for each uneven order, and, compounded of these, we have for the degree 2, and order 2e, e, or «+1 compounds; also of degree 2 and order 2e we have n—e+1 irreducible symbolic products. Hence « n—e,+1+ number of syzygies =eq+n—e+1, or =e +l+n—e+l1, according as ¢ is of the form 2e, or 2e, +1. In either case we find the number of syzygies equal to zero. 158 Mason MACMAHON, MEMOIR ON THE ORTHOGONAL AND SyYZYGIES OF DEGREE 2. Order of Quantic 2n. For a quantic of even order 2n the table of fundamental symbolic products is Order 0 1 2 3 4 5 6 7 8 0 1 | | | 1 1 1 1 1 1 3 SOs ieee ne = o i) n n—1 n—2 | a) n—4 | | 3 | dn? |4(n—1)? b(n — 2)" }(n—3)? 4 (n—4) | | where, in the fourth row, we take 3 {(n—e)—1} if n—e be uneven. Here, for degree 2, the number of asyzygetic symbolic products of type aPa"—8 2-2 is equal to the number of binary partitions of 2n+ €, no part being greater than 2n. This number is equal to n— «+1 or n—« according as ¢€ is equal to 2e, or 2e,+1. Moreover there is one irreducible product of degree 1 for each even order, and, compounded of these, we have for the degree 2 and order 2e, e, + 1 compounds ; also of degree 2 and order 2e there are n—e irreducible symbolic products. Hence, for ¢€= 26, number of syzygies = (e+ 1)+(n— 2e) —(n— 4, +1)=0, and for «= 2e,+ 1, number of syzygies =(¢&+1)+(m—2e—1)-—(n—«)=0. In both cases the number of syzygies is zero. Hence for no quantic is there a syzygy of degree 2. SyzyGIEs OF DEGREE 3. Order of Quantic 2n +1. I proceed to prove that the number of fundamental syzygies of order 2e+1 is the nearest integer to 4(e+1)*, where n>e. OTHER SPECIAL SYSTEMS OF INVARIANTS. 159 The number of asyzygetic symbolic products of degree and order 3, 2e+1 is, by previous work, (2n + 2)(2n + 3)(2n + 4) (1)(2)(3) Compounds are obtained by multiplying products of degree 1 and order 2¢+1 by products of degree 2 and order 2e— 2¢ for all values of ¢ from 0 to e; the number of these is Cosnteta / (n—e+1)4+(n—e)+...4+(n +1) =n(et+ 1) = (5) +1. Compounds are also obtained by taking the products of degree 1, three together; the number of these by a well-known theorem is equal to : 1 Co y(3)(8) Moreover the number of irreducible products is 4(n—e)(n —e€ +1). Hence the number of syzygies is 1 oe (2n + 2)(2n +3)(2n + 4) (1) (2)(3) (1)(2)(3) It is somewhat remarkable that the number, denoted by this expression, is independent of n and probably there is some easy way of establishing this a priort. I found this to be the case by actually working out the value of the number. The result is easily verified, so that I do not intend to reproduce the investigation here. mle 1) = (3) Pt eae Gp Del) Ca; COsne+2 Assuming the value to be independent of x it follows that mn may be assigned any suitable integer value for the purpose of arriving at the number of the syzygies. I find it most convenient to put m equal to e+1, viz, to assign to m the least value which yields an irreducible product of degree 3 and order 2e. The expression to be evaluated then becomes fone dele) eee Co ore = Coy aa a0) =te+5e+3+Co., ee — Cox.45 aia + Comma) ; When e=2e, the last three terms may be written Cox., seroe se) oo . ae Cose, + ees On = + Cog.,45 a ; or Co., aed — Cog. +8 os + 2Cos.,41 ap : or as _ Com aytray +2 es 3) : a Ca se ne 5a + ana ote 5 fare _) 160 Mason MACMAHON, MEMOIR ON THE ORTHOGONAL AND 5 (1+ 2a + 2a? + x) (264+ 2 or — Co, ry 2 i ) > (1)°(3) ( 2 1 1 2e,+2 . es er ss 9 1 or 5Co, a 5C@o._5 a + 2( 2 i: / 2 or a Ge aaa os are | Hence, for «= 2e¢,, the whole expression is Qe + 5e, + 3— $ (a2 + 3a, 4+ 2)— $ (Ee +e) + 4° + Ge, + 2, which is ¢2+¢, or $(e+1)—4. On the other hand, when the three terms may be written 1 2 Co., ae = C 8,49 77), . 8 (1)(2)(3)_****1)(2)8) ? i 1+2 N - 3 ee ———— . ow Co,,, (1)(2)(3) Co., (1)(3) 1 He 4 (1+ 2(1 +2) 8at+9 77\/0\/9\ se = =f SSS Conse Gyayiay 7 Cm CECB) ~ COM ORO) = (1+2) (1+ #)(1+ 2) = Cos..42 Aaa7e =C mat? /p/R\ on a(Ch TO) mae (2)°(6) a 14102 + 1022 + 1023 + x = 0.41 (1°(3) , Therefore 1 1 Coon Ty (ay(By ~ ®VE) 1+ 9a + 1022 + 927 4+ a =e yee) lg Py Amel 1 =— Co. ay 8Co,, ay Co, ay (Ser) fe.4+2 ea+1 Sees en, Hence for «=2e,+1 the whole expression is 4 (2e, +1)? + $ (264 1) +3 —F(e24+ 5a + 6)— 4 (a2 + 34,4 2)—F(e* +4) 1 at 200,41 (ys > which is — 3e2— 86, —5 + 4624+ 10. + 6=624 24 +1=(44+)), and this is d(e +1). OTHER SPECIAL SYSTEMS OF INVARIANTS. 161 Hence the number of syzygies of degree 3 and order 2e to the quantic or order 2n+1 is in every case the nearest integer to r(e+1), where, from previous work, e does not exceed n. As an illustration we have one syzygy deg-order 3, 3 for the cubic, viz.:— aya E>. By yiy?be? = ay ooryrys? . BE,°, one syzygy deg-order 3, 3 for the quintic, viz. :— a,5a,°E, .B Boye. 2 = OF y"72° Br BoE, and generally one syzygy deg-order 3, 3 for the 2n + 1-ie, viz.:— a,” Hare, 2 BeBe yes — APM Myott1 ’ [spe] of Ses Similarly for the quintic we have two syzygies of deg-order 3, 5, viz.:— ay"B, BE? . yiyobs = aE. 14728824, ara7&. By Py sh = BYE, . a%as%y,?7,*, and generally, for the 2n +1-ic, n>2, CRO A = op SoH Ghee B Yi yes =— Coes e Ss ‘ Wee ay eye yey sea a Hane, - [opener ewes lates = [Stan feymctet : ara Mey My tt ; the syzygies for the 2n + 3-ic being obtained from these by multiplication throughout by A, 2/3, Payirya- SYZYGIES OF DEGREE 3. Order of Quantic 2n. It will be shewn that the number of fundamental syzygies of order 2e is the nearest integer to te The number of asyzygetic symbolic products of degree and order 3, 2e is, by previous work, Co (2n + 1)(2n + 2)(2n + 3) mee (1) (2) (3) Compounds are obtained by multiplying products of degree 1 and order 2¢ by products of degree 2 and order 2e — 2¢ where ¢ has all values from 0 toe. The number of these is (n-e)+(n—e4+1)+...4+7 or n(etl)—(*$"). Compounds are also obtained by taking the products of degree 1 three together. The number of these is 1 Oo. Gaya): VOLe DOXe PART, 21 162 Mason MACMAHON, MEMOIR ON THE ORTHOGONAL AND Moreover the number of irreducible products of degree 3 and order 2e is L(n—e) or $(n—€)—4$ according as n—e is even or uneven. Hence when n—e is even the number of syzygies is (2n + 1)(2n + 2)(2n + 3) (1) (2) (3) It may be verified that this number is independent of x; we may therefore give n any value such that n—e is even; it is convenient to put n=e and then the number has the expression q 1 n (€ + 1) = de(e€ + 1) oF 3(n = ey + Co, (1) (2) (3) = Contre 1 1 5 eae Hele FD + CoM) “MG ) = Oe) whic 1S 1 1 de(e + 1) +e(e +1) + Co, M@2@)~ Co, (1) (2)(3)° | 1 _o, ite+2)_¢, 1+ Nos Com Vw" GO ~ AV) _q, A+e)(1+ 2.42%) a on ae an (1+ «)(1+ + 22°) Ee (13) 1 1 1—(1+#)(14+ «+ 22°) =e de GANA C € BENG 3a (14+ a+") 4+ 2a°(1+a+4 2°) ; (1) (2) (3) 1 1 = __—. — 2Co,.; => =< > 30.4 Gay — 20 GE) If «= 2e, this is : x(1l+a xy (1+2? — 3Co.,, QF > 2COse.—a ay an 22 1 +2 or = 3Co,, (1) = 2Co 0.4 a ; a 26 (e a ls a + ee =e) or bea eee giving a number of syzygies equal to $2e,(2e, +1) — 5e?— 3e,, which is e? or te. If «= 2e,+1, Gee ite: wl ape meee * (1)(2)(3) *(1)(2)(3) a (+ x) “4 a(1+ xp = —30C0.,, Oy 2Co.,, yr l+za uly — 3Co,, “ay — 4Co, + (ay 3 OTHER SPECIAL SYSTEMS OF INVARIANTS. which: is ae es *) - 3 (35 u *) - mn (* a ”). 2 2 2 or —5e?—8e,—3; giving a number of syzygies equal to $(2e, + 1)(26, + 2) — 5e,? — 8e,—3, or Grae Gis or te—4. Hence when n—e is even, the number of syzygies is the nearest integer to te. Next consider the case when n—e is uneven; the number of syzygies is then = =a Piale & be (2n + 1)(2n + 2)(2n + 8) n(et+1)—te(e+1) +43(n—-e) 4+ Co. Gay) Cone AOTE) Putting n=e+1, we find oe Jae re 1 1 ’ (e+ 1) — $e(e+1)+ Co, 2) Cots DG) + Co, OE or, after some simplification, : 1 d(e+1)(3e + n) + Co, Oe Coser =e ; elec a (1+ 2)(1 +2) M2) ~ @ye) e+e _g, 2 +2)(1 + 2) (1)°(3) ZENS (2) (6) Sag Ea Gee ae) " G¥@) 7 1 l Hence Co, Me@@)~ Cons NOTE 1—(1+2)(3 + 2x + 32”) (1)(2)(3) 2(14+a@+°)+3a(1+a+ 2°) (1)(2)(3) — 3Co,._, Now Cose45 = Cane = Cos.+2 = (Oho. = — Co, Ste (1°(2)° (142) 59, (bay QF 7 8% ey l+z 1 aay Oy 6 +2 at+1 at¢+l 5 Se aay acl.) or —5e?—Te—2 or —8@&—Ze—-2, = AO) ao O® (2) If ¢=2e, this is — 2Co,., or — 2Co., Hence the number of syzygies is (e+ 1)(Be+ 4) -—fe —Fe— 2, or dé, 163 164 Mason MACMAHON, ON ORTHOGONAL AND OTHER INVARIANTS. 1 1 Coe Ty(ayay ~ C+ GDB) a(1+a)y (1+a)y Qy 2 1 l+a « TW — BCO, 71\3 ? yey rake SON 2 fer . or =7(% Jeelaee: or — 5e?—12e,—7. a lif e=-een-t le = = 200r45 = 3Coe =— 4Co Hence the number of syzygies is d(e+1)(3e+ 4) — $(e —1— 6 (e-—1)—7, or fe— 4. Hence when n—e is uneven the number of syzygies is always the nearest integer to }é. To resume these results we have the theorem :— The number of syzygies between symbolic products of order 3 and order 2e+1 or 2e to the quantic of order 2n+1 or 2n, the products being free from &,, is the nearest integer to t(e+1) or te respectively. Doubling these numbers we obtain the number of syzygies, free from ££, amongst the co- variants. Illustrations are: one syzygy, degree 3 order 4 for the quartic, viz.:— onraak.? . Biiqys&e = evans - BE ; one syzygy, degree 3 order 4 for the sextic, viz. :— aysa1s"Eo" . Pa Bayn'rystEa? = antes *rys* « By Boks's two syzygies, degree 3 order 6 for the sextic, viz.:— mye Es . m°ByBtEst = BY Boys ye? . °E', Biya? Ea? « 1° O2Eo* = ay°aoryrye’ . Br°Es*. The corresponding syzygies for the quantic of order 2n are :— Degree 3 order 4, Gunes . [SRT ISH ay fo ee = CHEE Op ye : Sila Baca ea Degree 3 order 6, Yanan es 3 Ci aes Oita Oa = [Sher SeN oe aN , Gantsg see. [Sper Baten teeny eas E.? : CH E, = Cte Ore ey ey ea 2 jsyeer Copa So far the results are complete m regard to the third degree in the coefficients for all quantics. I have obtained incomplete results in regard to the fourth degree in the coefficients which I reserve for a future communication. VII. A comparison of the results from the Falmouth Declination and Horizontal Force Magnetographs on quiet days in years of Sun-spot maximum and minimum. (From the National Physical Laboratory.) By C. Curee, Sc.D., F.R.S. [Received and read February 12, 1906. ] CONTENTS. SECT. PAGE 1. Introduction . . . : é = c é 4 : 5 - - 165 2. Wolfer’s sun-spot frequencies . : : : . : : : : A . 166 3. Greenwich solar data - - : : - : : : > ; - Lay, 4, 5. Diurnal Inequalities. Ranges &e. . - : : 5 : s : ; - 168 6—10. Diurnal Inequalities. Vector Diagrams. : - : : . 5 , = S70) 11—15. Diurnal Inequalities. Fourier Coefficients . 3 : é . : : = lees 16, 17. Annual Variation. Fourier Coefficients . ; : 5 : > : : . 183 18—20. Wolf’s formula. Values of a and 6 constants. 4 c - ‘ . kets) 21—24. Comparison of results from Wolfer’s sun-spot frequencies and from Greenwich areas of whole spots, umbrae and faculae . . : : : - Ae kis) § 1. In a recent paper, described for shortness as (B)*, I have discussed the mean results from the Falmouth magnetic records for 1891 to 1902 on the “quiet” days selected annually by the Astronomer Royal. A previous paper described as (A)+ contained a similar discussion of the Kew results for 1890 to 1900, and in addition investigated the differences at Kew between a sun-spot maximum period 1892 to 1895 and a sun-spot minimum period consisting of 1890, 1899, and 1900. The present paper aims at investi- gating the ditference between a sun-spot maximum and a sun-spot minimum period at Falmouth. For the former period I have taken the same years as for Kew, viz. 1892 to 1895, but for the sun-spot minimum period the four consecutive years 1899 to 1902. It was shown in (B) that in the average year there is a remarkable agreement between a number of phenomena at Kew and Falmouth, especially the times of occurrence of maxima and minima. In (A) it was shown that a considerable difference exists between sun-spot maximum and minimum years at Kew, not merely in the amplitude but in other features of the diurnal inequalities, and it was obviously desirable to make sure that this was no purely local phenomenon. It is not of course safe to assume that the difference between two sub-groups of years from a single sun-spot cycle is an absolutely exact and unchanging measure of sun-spot influence. One 11-year period differs from another in its average sun-spot frequency. There is * Phil. Trans. Roy. Soc., Vol. 204, A, p. 373. + Ibid. Vol. 202, A, p. 335. WOlin, XOR Wa, WAM 22 166 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE also some direct evidence* that the relation between sun-spot frequency and magnetic diurnal variation varies somewhat from one epoch to another. Still it would appear that no great change in this relation has taken place since regular magnetic observations were commenced, and the sun-spot cycle now dealt with is a fairly average one, thus the results to which the investigation leads may be accepted with considerable confidence as fairly representative. Even if less representative than I believe them to be, they will serve a useful purpose in bringing out the notable differences that exist in magnetic phenomena within a short series of years. § 2. In (A), § 2 gave particulars of Prof. Wolfer’s sun-spot frequency from 1890 to 1900. The following Table I supplies the additional details necessary for dealing with the period 1891 to 1902. Wolfer’s great Tablet of revised frequencies ended with 1901. The data in Table I for 1902 are from his quarterly lists in the Met. Zeitschrift which are liable to revision. It is unusual, however, for revision to introduce more than minute changes in Prof. Wolfer’s figures, and as 1902 was a year of very small sun-spot frequency, revision could hardly modify the mean results to an appreciable extent. The differences in the last line of Table I are employed in calculating the constants of Wolfer’s sun-spot magnetic relation formula by the method of groups of years described in (A) § 52. TABLE I. Sun-Spot Frequency (after Wolfer). Jan. Feb. | March | April | May June | July | August | Sept. Oct. | | Mean Noy. Dee. | whole 0-7 1:0 0-6 3h) 38 0-0 (oe) 1901 02 | 2-4 4:5 0-0 | 10-2 5: 1902 6-7 0-0 | 14:8 0-0 | 35 ll 0-9 L855) S: Om een 6:3 | 08 Means for | 1891 to 1902 | 36-64 | 39:25 | 33-73 | 38-02 | 40°35 | 42-11 | 40:27 | 41-12 | 40-63 | 38-71 | 32-16 | 36-28 | 38:27 1892 ,, 1895 | 72:65 | 75°10 | 57:22 | 79-05 | 83-25 | 83-72 | 79-85 92:45 | 66:07 | 73-40 | 61:07 | 75°77 | 74:97 1899 ,, 1902 8:95 | 6:30 |11:50 | 7:55 | 9-15 9:87 | 5-85 | 2°50 | 6:32 | 10°57 | 5-60 | 2-90 | ” =| | Difference s S max.— S min. bone | | | | As has been remarked by Mr Ellist, when we consider the whole series of years for which Wolf’s and Wolfer’s figures exist, the mean value is so nearly alike for each season of the year as to discourage the idea of an appreciable annual variation in sun-spot frequency. But if we confine ourselves to 11, 22, or even 33 years, very considerable differences do exist between the means for different months of the year. In the present case it happens that certain months * Phil. Trans. Roy. Soc., Vol. 203, A, pp. 154, 158, &e. + Met. Zeitschrift, 1902, p. 193. + Monthly Notices, Royal Astronomical Society, Vol. 60, p. 153. 68°80 | 45-72 | 71:50 | 74-10 | 73°85 | 74:00 | 89-95 | 59-75 | 62°83 | 55-47 | 72°87 | 67-7: FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 167 show relatively low frequencies in years of sun-spot maximum and relatively high frequencies in years of sun-spot minimum, whilst other months have high relative frequencies in years of maximum and low frequencies in years of minimum. Thus the differences in the last line of Table I do vary somewhat conspicuously, the value for August being nearly double that for March. The uncertainties thus introduced are discussed in a recent paper*. §3. The present paper makes use to a minor extent of data for areas of faculae, umbrae and whole sun-spots as recorded photographically at Greenwich, Mauritius, and Dehra Din, and communicated by the Astronomer Royal to the Royal Astronomical Society. The figures employed here are taken from the Society’s Monthly Notices, Vol. ux11, Table I, p. 465. This Table gives two sets of data for the mean daily areas of the year, described here for brevity as “ Projected” and “Corrected,” or “P” and “C.’ The former are the areas as seen and measured on photographs, expressed as millionths of the sun’s apparent disc; the latter are the areas as corrected for foreshortening, expressed as millionths of the sun’s visible hemisphere. For the present purpose the size of the unit employed in any measurement of solar activity is im- material, but it is desirable to compare the modes of variation of the several quantities throughout the 12 years. This object is secured in Table II by expressing the faculae, umbrae or whole spots, as the case may be, in terms of their mean value throughout the 12-year period taken as unity. The absolute values of the several 12-year means, in the units employed by the Astronomer Royal, are given at the foot of the Table. Comparative data are also given for Wolfer’s frequencies. Table II shows that whilst the mean absolute P and C values differ by from 10 to 40 per cent. according to the element, their variation proceeds in each case in a very similar way throughout the twelve years. We thus see at once that so far as concerns the existence of a general connection between magnetic elements and the Greenwich faculae, umbrae, or whole spots, the conclusion come to is sure to be pretty much the same whether use is made of the projected or corrected areas. It is also evident that the mode of variation in the areas of whole spots and umbrae is so closely alike that it is sure to be pretty nearly indifferent whether we employ whole spots or umbrae in Wolf’s formula in place of his sun-spot frequency. It will be noticed that the means of the corrected areas for whole spots and umbrae are as nearly as possible in the ratio of 6 tol. If this somewhat curious tact be borne in mind, the parallelism in the variation of whole spots and umbrae is readily recognised even in the Astronomer Royal’s absolute values. The mode of variation of the faculae is conspicuously different from that of whole spots or umbrae in the years of sun-spot maximum, there being an enormous excess in the figures for 1892 as compared to subsequent years, especially 1894, which is not seen in the case of the other elements. * Phil. Trans. Roy. Soc., Vol. 203, A, p. 151 (especially § 18, p. 166). 22—2 168 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE TABLE II. Mean annual values of Faculae, Umbrae, and Sun-spots, expressed as decimals of their arithmetic mean from 1891—1902. GREENWICH | z : | Year Faculae Umbrae | Whole Sun-spots | Pikes 2 Brojeated || \uCormctea! Merete Cerne | Projected | Cneted | | | 1891 1120 | 1:100 0-850 | 0-861 | 0-918 0-947 | 0-930 1892 2°735 2-546 1:827 1857 | 1:966 2-020 1:907 1893 1:937 1:872 2734-35) eeoDe | 2-443 | 2-436 2-218 1894 1-411 1-462 | 2:271 | 2-306 2°129 2-133 2-038 1895 1°744 1-774 1-698 | 1-687 1639 1621 1:672 1896 | 1:053 | 1:098 | 0-910 | 0-898 0-918 | 0-903 | 1:092 | 1897 | 0:827 0-895 | 0874 | 0879 | 0856 | 0-855 | 0-685 1898 | 0-650 | 0-694 0-666 | 0639 | 0°655 0-624 0-698 1899 0:252 | 0-262 | 0-193 | 0-180 0196 | 0-185 | 0-316 1900 | 0:127 0-140 0-158 0-170 0125 0-125 0-248 190] 0-019 0-023 | 0-100 0-086 0-051 0-048 0-071 1902 0-127 0-134 | 0-100 0-100 0-106 0-103 0-123 Mean | | crea 1180°9 ieee | 139-58 | 100-15 | 811-75 | 601-00 38:27 DiURNAL INEQUALITIES. RANGES, WC. § 4. Table III gives for both Declination and Horizontal Force—referred to generally as D and H for brevity—the ranges and the sum of the 24 hourly differences from the mean of the day in the diurnal inequalities for each month of the year and the year as a whole. “Range” here means the difference between the greatest and least mean hourly values. There is a considerable “non-cyclic” or aperiodic element in the annually published Falmouth Tables, especially in H (see (B), p. 376), which has been eliminated in forming the diurnal inequalities. The quiet days numbered five a month, so that an hourly mean for a particular month from a four-year period is based on 20 curve measurements. Individual measurements aim at 0°1 in D and 1y in H, so that the degree of accuracy in the hourly values should be of FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 169 the order 0°01 in D, and 0'1lyin H. For convenience in eliminating the non-cyclic element the hourly values were really taken out to 0:01, but the last figure has been discarded in Table III in all the H columns, as well as in the D columns devoted to the sum of the 24 differences. The discarded figures were used, however, in calculating the results dealt with later in Table XV. TaBLeE III. Declination (unit 1’) Horizontal Force (unit 0-1) Month Ranges Sum 24 differences Ranges | Sum 24 differences Smax. | Smin. | Smax. | Smin. | Smax. | Smin. | Smax. | Smin. Tanne St BOR eS e Sis | 16-98 |) /251 128 | 1970 | 561 Webrdaxy 671 | 377 | 430 | 189 | 296 116 | 1620 | 702 Merealnes 61 10-74 | 726 | 517 | 330 | 368 207 | 2289 | 1087 pals 1227 | 920 | 568 | 399 | 491 316 | 3017 | 1752 Maye 1199 | 9832 | 598 | 363 | 509 74 | 3340 | 1755 Fane se. 1242 | 859 | 652 | 432 | 513 | 328 | 3979 | 1989 cain escccied 1218 | 880 | 602 | 410 | 505 | 312 | 3286 | 1806 August ....... 1279 | 893 | 627 | 495 | 495 | 309 | 3146 | 1750 tani eiias. | gon, | et | 372 | 423 284 | 2453 | 1635 @rtcber 0. 870 | 647 | 463 | 294 | 345 | 248 | o195 | 1540 November... 601 385 | 313] 168 | 278 155 | 1738 | 881 December ...| 426 | 224 | 280 | 127 | 191 71 | 1071 416 | | Year......| 947 | 625 | 496 | 302 | 373 | 219 | 2958 | 1236 The range as defined here is necessarily an underestimate unless both maximum and minimum should happen to coincide with exact hours G.M.t. The sum of the 24 differences is less dependent on the choice of the origin of time, and is probably the better measure of the activity of the forces to which the diurnal inequality is due. It will be noticed that whilst December is unquestionably the month when the diurnal inequality is least, the difference between December and January is less marked in the case of the 24 differences than in the case of the ranges. It is also less marked in the sun-spot maximum than in the sun-spot minimum period. § 5. The relative variability of the diurnal inequality throughout the year is, as at Kew, decidedly less in years of many than in years of few sun-spots. This will be readily seen on 170 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE comparing the two curves of Fig. 1. The full line curve answers to the sun-spot maximum, the broken line curve to the sun-spot minimum group of years. The ordinates represent JAN. FEB. MAR. APRIL MAY JUNE JULY AuG. SER Oct. Nov. DEc. JAN. Fie. 1. the ratios borne by the sum of the 24 hourly differences for individual months, as given in Table III, to their arithmetic mean. The ratios shown are the means of those calculated for D and H. § 6. A complete record of the mean hourly values in the diurnal inequalities for each month of the year in sun-spot maximum and sun-spot minimum years would occupy too much space. A good general idea of the phenomena can be derived from the accompanying vector diagrams for December, March, June, and the year as a whole (Figs. 2 to 5), The dotted curves are for the sun-spot maximum, the full line for the sun-spot minimum period. As already mentioned, December represents the minimum of diurnal movement. March is fairly representative of the equinoctial months (March, April, September, October), and June represents the four summer months (May to August). MS and EW represent astronomical north-south and east-west directions. Their intersection serves as origin, and the line MM’ represents the mean position of the magnetic meridian during the 12-year period, viz. 18° 47-4 west of north. The numerals indicate the hours of the day, counted from midnight G.m.r, The line drawn from the origin to any numeral represents in direction and magnitude the value at the hour specified of the horizontal component of the disturbing force to which the diurnal inequality may be ascribed. The scales on which the curves are drawn are indicated on the diagrams (10 y = ‘0001 c.G.s.). The sun-spot maximum and sun-spot minimum curves are drawn from a common origin except in the case of December. The difference between the size of the vectors in years of S max. and S min, is too obvious to require detailed comment. FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 171 DECEMBER, 10¢y _ 107 x9 172 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE JUNE. FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 173 § 7. The mean declination for the S max. period really exceeded that for the S min. period by 3675, but so small an angle could not have been shown clearly in the diagrams. It is obvious to the eye that the vector for the S min. curve is in advance of that for the S max. curve at noon and in the early afternoon, but this is not generally the case throughout the whole 24 hours. Owing however to the slowness of the angular change throughout the night, vector diagrams would require to be drawn on an excessively large scale to facilitate a study of this phenomenon. I have accordingly calculated the exact hourly values of the vectorial angles in both the S max. and S min. periods. The approach of the S min. period to an ideal definite condition, viz. a total absence of sun-spots, would naturally lead one to pay it chief attention; but owing to the smallness of the diurnal changes, the calculated vectorial angles for this group of years are exposed to considerably larger probable errors than those for the S max. group of years. The latter have thus been selected for exhibition in Table IV. The angles, though recorded only to whole degrees, were calculated to the nearest 01. They are TABLE IV. Vectorial Angles for S max. period. Forenoon Hour...... 1 | 2 3 4 5 6 7 8 9 10 1l 12 January... 48 | 38 | 27 | 20} A NeelGialesh tlewnit | 34 172. 202 February..| 52 | 58 | 62 | 51 | 43 | 40 | 35 | 45 | 84 | 132 | 173 | 203 March - | 2a -| 19 | 20 | 23°) 98 | $37 49 | 75 | 99 | 126 | 164 | 201 April <..!. 36 15 | 26 | 32 | 35 | 42 | 62 | 78 | 99 | 125 | 157 | 194 May ....... lee ele rset Ae con, | eal 86 103) 118) | 142) |) 172) 2038 June ....... 10 Pe ot | 27, | OY | aa Ole 03 06 | 137° e1Gees e206 aly ee ere Teel gael Scot 5b, |) Ae OG tia: | ler lem menos August Seal eae tien Shea) 750, | Goa Ise ATOM /et200 ||) 147. srSOns moa September 18 17 29 33 44 59 79 98 | 121 || 150 | 83 ieoon October...) 26 | 22 | 18.| 15 Vie 24 STi Ta LOS || 1295 Sth mens November) 25 12a ate 5 Over) 16m eee eal 97 | 131 | 165 | 197 December | 77 | 76 | 55 | 29 | 22 | 20 | 93 | 41 Ti || 1295 elem 205 | | Vea ot Dome eZOne «Soe eat 527) 68 "| “s6 |/108 |)’ 136 io 205 Wits, OX ING, WA0E 23 174 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE TasBLE IV continued. Afternoon Hour...... 1 2 3 4 5 6 7 8 | 9 10 11 12 January...| 231 | 240 | 233 | 238 | 299 | 274 | 334 | 378 | 399 | 409 | 4i0 | 413 February..| 221 | 233 | 239 | 244 | 255 | 305 | 321 | 344 | 364 | 385 | 397 | 402 March ..... p93" 937 ||240" | 261!) "276""| 323!) 352" | 358 | "S689 (0873) | "372 eee April ...... 215 | 231 | 246 | 268° | 289 | 327 | 340 | 339 | 343 | 341 | 351 | 361 May ....... 225 | 239 | 255 | 280 | 308 | 328 | 340 | 340 | 345 | 342 | 348 | 356 June ......<| 227 | 241 | 258 | 975 | 294 .| 313 | 332 | 3387 | 341 | 337 | 350 | Bbe July ....... 219 | 236 | 254 | 270 | 300 | 324 | 337 | 340 | 342 | 340 | 345 | 354 August ...| 232 | 244 | 955 | 975 | 314 | 346 | 347 | 348 | 351 | 354 | 357 | 368 September) 239 | 248 | 252 253 | 293 | 349 | 352 | 358 | 359 | 361 | 365 | 374 October ...| 223 | 233 | 242 | 240 | 263 | 315 | 340 | 349 | 369 | 376 | 384 | 386 November| 213 | 227 | 235 | 247 | 286 | 310 | 328 | 353 | 375 | 390 | 401 | 401 December | 220 | 228 | 234 | 241 | 271 | 309 | 336 | 366 | 387 | 408 | 413 | 430 Year...| 224 | 237 | 248 | 261 | 290 | 323 | 340 347 | 355 | 360 | 369 | 377 measured from an initial line drawn towards astronomical north, the positive direction of rotation being supposed clock-wise (N., E., S., W.). To avoid the appearance of discontinuity, angles in excess of 360° usually appear in the late afternoon hours, but the meaning attaching to, say, 400° is of course the same as that attaching to 40°. The Table will, it is hoped, afford a useful check on speculations as to the immediate cause of the diurnal inequality. In any comparisons it is important to remember that the hours being G.M.T. differ considerably—especially in months when the equation of time is large and positive—from local solar time*. The variation in the vectorial angle with the season will be seen to be com- paratively small from 10am. to 4p.m.; it is large from 10pm. to 2am, and also at 6, 7 and 8am. and at 5 and 6 p.m. § 8. To give full details of the differences between the S max. and S min. vectorial angles for every hour of every month would have entailed another long Table; accordingly in Table V I give only mean results for the four months comprising Winter, Equinox and Summer, and results for the year. The last, it should be noticed, are derived directly from the mean annual diurnal inequality, and so, unlike those for the three seasons, are not arithmetic means of differences for individual months. The sign has been taken as plus * Falmouth local time is 20-3 minutes later than Greenwich. FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 175 TABLE V. Excess of Vectorial Angle in S min. as compared to S max. period. Forenoon Hour ...|/ 1 “| 2 awe 5 6 7 | 8 9 10 Ty Sipps s = boll | Winter [ret | +6} =651| =a01} Say | 13 | —3 | Speen | eis 1s ed Equinox | — 4 | +2 O/;/+2/+2/+ 3] —-1 —2 —-2 ;-2/+ 4/4 5 Steet Sash oda nerr aroel a kG teat sd (ta alain Year — 5 -5 |}-9/]-—-5]/-1]+4+ 3] +2 —1 —1 OW peas iheeas Afternoon | Hour...| 1 2a g) GRE 4 | 5 6 rims NU: 9 10 u 12 Winter | +14 | +11 +9 | + 8 | +24 | +59 | +58 | +30 | +22 | +11 | +16 | +15 | Equinox | + 8 Ps 8 | +7 | +12 | +40 | +22 |4+12 + 9/+7 | + 6 | + 8 cea errenars |44°9. (4 +1 )/-5j)- 1 te TS ic Oes eet + 3 | + 3 OS | ea 410 Vane | oe ea id Near ere ieee he @ |e | + B25 when the vectorial angle is larger for the S min. than the S max. group of years. A query is attached to the 3 and 4am. data in Winter. At these hours the February S min. vectorial angle presented outstanding values, and owing to the minute size of the vector itself much might have been due to an element of chance. It will at once be seen that the differences between the S max. and S min. vectorial angles are systematic and have a well marked diurnal variation. On the average of the 24 hours the S min. angle is the larger, by 8°75 in Winter, 6°0 in Equinox, 1°3 in Summer, and by 4°1 in the mean diurnal inequality for the year. The excess of the S min. angle is most marked about 6pm. and 1 pm.; from midnight to 4 or 5am. the S max. angle is usually in advance. § 9. The times during which the S min. and S max. angles are respectively the larger are shown graphically in Fig. 6 for the twelve months individually and for the year as a whole. The line is existent or non-existent according as the S min. or the S max. vectorial angle is the larger. The transition time is calculated in each case by supposing the change from plus to minus or from minus to plus to take place uniformly throughout the hour during which it occurs. In one month, August, the S min. angle was the larger throughout the whole 24 hours, but at one hour the excess was only 1° so the phenomenon is probably in part accidental. 23—2 176 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE On the average of the 12 months the S min. angle was the larger during 16, and the S max. during 8 hours of the day. This is practically the same proportion as in the mean diurnal inequality for the year. January February ——_—— March April bn May ee June ee | July ——— oo i August September SS SS October a November December Year § 10. From some points of view the rates of change of the vectorial angle are of more interest than its absolute values. In the case of the S max. group of years the hourly increments in individual months are immediately derivable from Table IV. As a comparison with S min, years appeared however desirable, I show side by side in Table VI, for the two groups of years, mean hourly increments of the vectorial angles for the three seasons and for the year. The values for the seasons are arithmetic means of the values for the individual months; those for the year are derived from the mean diurnal inequality for the year. The S min. increment from 3 to 4am. in winter is queried for reasons already explained. As one would have anticipated from the remarks in § 7, the hourly increments are markedly less regular in the S min. than the S max. period; but in addition to irregularities ascribable to observational uncertainties, there is in the S min. period an unmistakable tendency to a greater difference between the fastest and slowest (or most retrogressive) hourly increments than in the S max. period. The phenomenon, I may add, is even more distinctly shown when one considers the individual months. If the difference between the algebraically greatest and least hourly increments be defined as the range, then the mean value of the ratios borne by the S min. to the S max. ranges from the 12 months of the year exceeded 1°8, the S min. range being the smaller in only one month, June. FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 177 TABLE VI. Hourly increments in Vectorial Angles. Forenoon | ] ours -<-2-2- seg see Pe ere) pats 5—6 | 6—7 | 7—8 | 8—9 | 9—10 | 10—11 aa Ue | | Winter (2X | —1 |- 4)-7 |-18|-4)- 3] 4+ 5 +19 | +40 | +48 | +40 | +30 veer [Smin. | +6 4—19 | —S1t] +15 | ¥ 9 | 413 | +15 | +22] +48 fae +41 | +28 (Smax. | +3 @ ee seh Sue 42 bee bald | 228 lea 24 | 2 970-) 4 35] 9 87 Equinox) ¢ min O;+ 7/+ 2]+ 5 | + 4/4 9/415 | +22 | +24 | +28 | +41 | +39 = (S max +9 | +11 | +11 13 | +15 | +16 +15 | +15 +16 | +24 | +30 | +34 Me (Smin, | +7 | +13 | +10 | +17 | +16 | +17 12 | +11 | +16 | +25 | +35 | +42 Y S max +4 fe 2E |p te gh |) ae 2h | aes |} +11 | +16 | seas |p see | sects |) eee) || Bhs aoe USmnin,. | l=] 4: & O}+ 8 | +12 | +15 | +15 | +15 | +22 | +29 | +38 | +39 | | | | | | | Afternoon | Hour ......... og | ep a se |e ee [as Meet G7) |i e281] o29e | 080: oan eae | | Winter (S max. | +17 | +13 | + 3 | + 5 | +20 | +39 | +30 | +31 | +21 | +17 | + 7 +6 \Smin. | +18} +11 | + 1] + 4/436 | +75 | +29 |4+ 3/413} + 5 | +13 | +4 tee fSinax. | 20°) +72) +100) 7° | 96 | ade) + 18h) ep aio | oe Per be | Mee Equinox{'¢ Min eoeen ioe hey See asim see SON etn Tole Gell a) Ds |e. ll eae | Sone |e: |e Ital eda lke |e ae ry | ef a4 l= a] 4s 7 | ee Geet (Sune | 20 | 11 | ED eds" | 433° 1 36 By eae po +4 Year....S max. | +19 | +13 | +11 | +13 | +29) +33 | +17|+ 7/+ 8|+5/+9) +8 US min = il saa la Wa +38 | +40 UR ead be o +2 |+ 8 ra DIuRNAL INEQUALITY. FOURIER COEFFICIENTS. § 11. The diurnal inequality in D or H can be expressed with high accuracy by four terms of a Fourier series whose periods are respectively 24, 12, S and 6 hours. We may suppose the result expressed in the form ¢, sin (f+ a) + cz sin (2¢+ a) +c, sin (3¢+ a3) + ¢, sin (4¢ + a4), the cs and a’s representing the amplitudes and phase angles, and ¢ being time counted from midnight (G.M.T.), an hour being taken as equivalent to 15°. The suffix , answers to the longest period, 24 hours. The values of the c’s and a’s for each month of the year are given in Tables VII and VIII for the S max. and S min. periods already indicated. The calculations were carried further than appears in the Tables, the phase angles being worked 178 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE TaBLE VII. Declination. Fourier Coefficients, Amplitudes and Phase Angles. Cy | es & Ci ay ay as a Month ——| - ~ = ——— Max. | Min. | Max. | Min. | Max. |} Min. | Max. | Min. | Max. | Min. | Max. | Min. | Max. | Min. | Max. | Min January...) 1°75 | 0°76 | 1°12 | 0-82 | 0-56 | 0:40 | 0-36 | 0-21 | 234 | 243 | 16 | 94 | 235 | 232 | 37 | 45 February..| 2°54 | 1-04 | 1-28 | 0-85 | 0-81 | 0-46 | 0-40 | 0-22 | 227 | 245 | 23 33 | 214 | 228 | 21 | 21 March ..... 2°63 | 1°70 | 2-49 | 1°58 | 1-41 | 1:10 | 0:56 | 0°53 | 216 | 222 | 25 | 33 | 209 | 211 |) 34 | 33 April ...... 3°22 | 2:05 | 2-71 | 2°18 | 1°58 | 1:21 | 0-43 0:40 | 204 | 210 | 27 27 | 207 | 210 | 38 | 37 WEN enascue 3°51 | 2:03 | 2-85 | 2:08 | 1:13 | 0-90 | 0:13 | 0:20 | 206 | 203 | 39 | 39 | 230 | 230) 66 | 51 JUNE ......- 3°95 | 2°53 | 2°92 | 2°15 | 0-96 | 0-81 0-03 0-12 | 201 | 204 | 34 35 | 231 | 223 | 102 | 22 Uy poeta 3°62 | 2°42 | 2°82 | 2:07 | 1:05 | 0:92 | 0:01 | 0-23 | 203 | 203 | 34 | 35 | 219 | 215 | 2612) -8 August....| 3°50 | 2-29 | 3:05 | 2:16 | 1-41 | 1:07 | 0:32 | 0-17 | 217 | 222 | 46 | 50 | 230 | 234] 44 | 34 | September] 3:11 | 1:91 | 2-66 | 1-82 | 1-37 | 1:05 | 0°39 | 0:35 | 224 | 226 | 44 | 45 | 230 | 231 | 55 | 55 October ...| 2°51 | 1:47 | 1:98 | 1-46 | 1-19 | 0-94 | 0:53 | 0-48 | 221 | 295 | 22 30 | 213 | 220] 30 | 41 November | 1:62 | 0-74 | 1-37 | 0-91 | 0-66 | 0-57 | 0-44 | 0-31 | 223 | 248 | 17 33 | 226 | 241 | 38 59 December | 1:55 | 0°64 | 1-06 | 0-64 | 0-43 | 0-30 | 0-25 | 0-12 | 234 | 266 | 17 | 31 | 230 | 235 | 30 | 44 | out to minutes of arc, and use is made of the more complete values later in the paper. If Falmouth solar time were used the phase angles would require the corrections given in (A) Table XIX, with the constant addition of +4° 45°6 in the case of a,, +9° 31’ in the case of a, and so on, Falmouth local time being 19 minutes 3 seconds after Kew (20 minutes 18 seconds after Greenwich). All the c’s, both for D and H, show a well marked winter minimum, nearly always in December. In c, and c, there is also a well marked summer minimum, in some cases the principal minimum of the year. The excess of the sun-spot maximum over the sun-spot minimum value is always large in ¢, and ¢, especially the former. It is less marked though generally apparent in ¢;. In c, it is not clearly shown except in winter. In the case of D the phase angles a, a), @ are almost invariably smaller in S max. This indicates a later hour for the daily maximum in the former In the case of a, in D, than in S min. years. than in the latter epoch. The difference is most distinct in winter. and all the phase angles in H, the difference between S max. and S min. years is not FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 179 sufficiently prominent to make its presence certain in individual months, when data from only one sun-spot cycle are available. TABLE VIII. Horizontal Force. Amplitudes and Phase Angles (Unit for c’s 1 ¥). fe C2 Cs | C | a, Oy a, a, Month | — : a ~ —_-— Max.| Min. | Max.| Min. | Max.| Min. Peieees femernial evens lone) Mix: I RGas | oxtac|| atin | Max. | Min | | |. i | (a | January... 68 | 24| 51| 30 | 24/18] 19| 12] 71] 7 | 264 | 289 | 137 | 185 |-93|- 3 February... 90 | 34 | 67) 28) 28/13/21 | 10] 95] 75 | 252 | 251 | 125 | 145 | -25|- 9 March .....|136 | 61| 80| 43 | 42 | 29] 16| 15 98 | 103 | 281 | 284 | 141 | 138 ~ 25 | 20 April ......| 18:3 | 10-3 |10-0 | 6-7 | 4-4 | 3-7 | 16 | 21 | 105 | 108 | 278 | 291 | 124 | 126 — 25 | —33 May........ jai li14| 96 | 40 | 26 | 4 | 1-0 | 0:8 | 125 | 124 | 298 | 305 | 155 | 194 | + 70| +33 June .......) 207 jio7 | 84| 54 | 24 | 19 | 0-2 | 0-8 | 128 | 119 | 303 | 288 | 187 | 173 |+85/—- 8 Fuly (20° [11-6 | 9-4 | 4-7 20 28 | 06 | 11 | 120 | 121 | 295 | 302 | 139 | 171 | +17] —30 August...) 19°5 [11-3 | 9-1 | 44 | 36 | 3-3 | 1-4 | 1-5 | 118 | 126 | 307 | 334 | 175 | 200 |+ 4/+16 | September! 16-0 |10-4| 63 | 49 | 47 | 3-2 | 2-3 | 1-7 | 113 | 117 | 313 | 320 | 188 | 180 | +21] +19| | October ... 13-0 91| 73| 55 | 3-4 "50 | 19 | 1:5 | 92] 95 | 280 | 284 | 151 | 146 | - Ais 3) | November 93 | a7| 73| 42 | 27 19 | 15 0-9 | 90 | 105 270 284 | 131 154 |—14] + 3 | December 5:0) 12) 5:9 | 25 | 14) 07 | 10) 03) 87 | 97 253 | 971 | 132 | 165 | -17 | -21| § 12. Table IX gives the values of the ¢ coefficients in the mean diurnal inequalities for winter, equinox and summer in D and 4H, and also in the mean diurnal inequality for the whole year in these elements and in W and WN (the westerly and northerly components of the horizontal force). The differences between S max. and S min. years are similar to those at Kew described in (A). The general nature of the difference is perhaps more readily recognised from Table X, showing the ratios borne by cs, c; and c, to c, in the case of the mean diurnal inequality for the year. The shorter period terms diminish in importance as compared to the 24-hour term as sun-spots increase. This is shown more prominently in D and W than in H and NV. Similar phenomena are shown in Table XXXVII of (A) at Kew. § 13. Table XI records the phase angles for the seasons and the year corresponding to the amplitudes appearing in Table IX; it also gives the algebraical excess of the angles for the S min. years over those for the S max. years. The tendency for this difference to be 180 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE positive, z.e. for the daily maxima to occur earlier in S min. than in S max. years, is much more marked in winter than in summer, TABLE IX. Amplitudes (Unit for c's in H, W, and N, 14). D H lee f NV Winter | Equinox| Summer | Year Winter | Equinox|Summer| Year | Year Year | Max. ...| 1°86 | 2:84 | 3-62 | 2-74 7-4 | 15:1 | 20 13-9 | 136 | 15-1 TW bt sonal) Werte) 177 2°30 D7 2°8 8-9 Maley 76 LET 8-4 Maxey. e||, Ls2i 2:43 2:90 PG || GW 77 9-1 7:3 10°6 88 “2 |Min.....| 0-80 IY (3) 2:10 155 | 3-1 5:2 4-4 AOS Aly MaV(SCS 52 | (Max. 0-61 e377) 103 EOS sma 3:8 2°6 2°8 57 27 3 (Min. ....| 0-43 1:06 0:92 O80 ples 3-0 2°3 2:1 4-4 169 (Max. . 0°36 0-47 O11 0:31 16 18 0-7 1:3 i) 0-9 =Mimnt ee.) 0-211 0:43 0-16 O27 || 910-9 16 1:0 11 16 08 TABLE X. Ratios of Amplitudes in mean diurnal inequality for the year. D A | W N | S max. Smin. | S max. S min. S max. S min. S max. S min. al | Co/ey oe ‘79 99 D3 “BA ‘78 | 1:00 D8 62 alerts | 1288 51 20 28 42 | +58 18 22 Wes enter lil aly “09 “ld 14 21 06 10 Comparing Table XI with Table XXXIX of (A) it will be seen that the differences shown in the case of a, for the mean diurnal inequality for the year are closely alike at Falmouth and Kew in all the elements. In the case of a and a, the Kew differences are decidedly the larger; whilst in the case of a, the results are somewhat indecisive at both stations. FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 181 TaBLE XI. Phase Angles. D H W N Winter Equinox)/Summer| Year | Winter | Equinox Summer) Year Year Year (Max. ... 22914 216 6 | 206 37/21452| 8636| 10242 12253|/10934/19620 9151 Ee | Min. ....| 249 23 | 220 0} 207 54 | 218 49| 88 40 | 106 33 | 122 36 | 112 29 | 20056 | 94 23 Min.— Max. 20 9 3 54 Lai | 357 2 4 51 — 017 2 55 4 36 2 32 | Max....| 18 8] 2952 | 38 27 | 31 31 | 260 0 | 285 57 300 35 | 284.52| 1912 26043 “2|Min.....| 3015] 3341] 40 5| 3559 | 270 39 | 294 27| 306 7 | 292 52) 26 22 | 262 40 Min.— Max. 12 7 3 49 | 1 38 | 4 28 | 1039} 830 5 32 8 0 7 10 Bye Max. ...| 222 23 | 214 27 | 227 28 | 220 46 1131 0] 151 31} 165 59| 150 19| 212 4/110 56 se (UMin. ....) 234 28 | 217 24 | 226 3 | 223 42 | 153 52 | 146 37| 185 2 161 6) 215 48 119 24 Min.—Max.| 12 5| 257) —1 25 256) 2252)—454| 19 3] 1047 3 44 8 28 Max. ...| 3131] 3816 | d3 8) 37 25 — 20 36 |— 5 33 | +30 26 | —551 2856 —30 14 “4|Min.....| 43 27) 4024] 2350] 3754|- 5 6|-1048|+ 210|—542| 29 5|—28 59 Min.— Max. 11 56 2 8 |- 29 18 029) 15 30|- 515 |-28 16 ORS UY 15 | | Taking means for the year from D, H, N and W, and converting differences of angle into time, we find for the retardation in minutes in the S max. as compared with the S min. period the following values: | 24-hour wave | 12-hour waye| 8-hour wave | 6-hour wave | | At Falmouth 140 108 | 85 | O85 | At Kew...... 15°5 185 | 14-0 | -01 The retardation though apparently less at Falmouth than at Kew is well marked in the 24, 12 and 8-hour waves. In the 6-hour wave it is doubtful whether the phase angle is affected in the mean diurnal inequality for the year; there is a decided retardation during sun-spot maximum in winter, but an acceleration apparently in summer. § 14. Supposing the diurnal inequality in years of sun-spot maximum to be composed of a fundamental part representing the inequality in the total absence of sun spots, and of a second part associated with sun spots, it is important to know whether the two parts are or are not essentially different in type. A conspicuous difference in type would naturally suggest a difference in source, while close agreement in type would suggest a mere intensification of action in years of sun-spot maximum. Worle exe Nos VLE. 24 182 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE Subtracting the hourly values for the sun-spot minimum period from the corresponding values for the sun-spot maximum period, we obtain a diurnal inequality which should be wholly free from what was called above the fundamental part of the real diurnal inequality. Fourier coefficients were calculated for this difference imequality for the three seasons and the year, and a summary is given in Table XII. TaBLeE XII. Difference Inequality (Diurnal) S max.—S min. (Units yor c's 1’ in Declination, ly in Horizontal Force.) Declination Horizontal Force Winter | Equinox | Summer | Year | Winter | Equinox | Summer | Year aaa g Oe Abe 1-28 1-08 1:33 | 118 | 46 63 | 9-0 6:3 Cane 0-45 0-70 0-80 063 | 32 | 26 | aa Saas | raion ie 0-21 0-31 022 0-24 | 1-2 os | 08 0-8 C,...-.| 0-16 0-04 0-09 00511] 150-67 0-38 Oi” iota f8) ilael Per ees 210 204 210 8 | 97 122 106 ce ay 20 34 2 | 250 | 269 | 995 275 | ae eee 197 204 | 233 211 103169 | 103) || tei | Thee 17 | 15 164 35 ea|ea38 | 36 | 141 7 § 15. In the S min. period the mean sun-spot frequency was only 7:25. Thus if the diurnal inequality really consists of a fundamental part and a sun-spot contribution, the phase angles for the S min. years in Table XI cannot differ very much from those of the fundamental part. Consequently if the sources of the fundamental part and the sun-spot contribution were wholly different we should expect notable differences between the angles in Table XII and the corresponding angles for the S min. years in Table XI. In two cases—the summer values of a, in D and H—there is a conspicuous difference, but in all other cases the differences are hardly such as to suggest a radically different source. Even the S min. angles in Table XI are slightly modified by sun-spot frequency. In the case of # and a, by assuming the difference in phase angle proportional to difference in sun-spot frequency, we can calculate angles for the fundamental part alone. The assumption is probably not exactly true, but the results can hardly be much in error. The phase angles thus calculated exceed those for the pure sun-spot element, whose approximate values are given in Table XII, by the following amounts: FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 183 | a, a | z | 4 Winter | Equinox | Summer | Year | Winter Equinox Summer Year | | } } | } ji van a2 | 129 | 37 96 | 35-0 | 140 | 61 | 160 ts ee 35 9-9 0-5 6-8 2146 | 268 US Ge ss | | | Mean 159 11-4 21 [ass 28°3 20-4 8-6 17-4 The mean differences represent a retardation in the sun-spot contribution as compared to the fundamental part, which in the case of the mean diurnal inequality for the year represents fully half an hour in both the 24 and 12 hour waves. The summer values of a, in D and H in Table XII already alluded to differ from the corresponding S min. angles by about 140°, so that there is here an approach to a reversal of phase. The phenomenon appearing remarkable, I calculated the values of a, at Kew corresponding to those in Table XII. They exceeded the Kew S min. angles by 226° in D and 144° in H. The amplitudes corresponding to the phase angles a, in Table XII are so small that accident may play a considerable part in the result; but the coincidence certainly points to something more than mere accident. ANNUAL VARIATION, FOURIER COEFFICIENTS. § 16. In (A) and (B) the annual variations of the range and the sum of the 24 differences from the mean, in the diurnal inequality, and likewise of the amplitudes of the 24-hour and other contributory waves, were shown by means of Fourier series with annual and semi- annual terms. In (B), § 16, attention was called to the remarkable resemblance between results for the average year at Kew and Falmouth. It was thus important to ascertain whether the results varied at all with sun-spot frequency. Accordingly Fourier series of the type M+ P,sin (+ @,) + P, sin (2¢ + 02) were calculated for the S max. and S min. periods separately, and the results appear in Table XIII. M denotes the mean value of the element throughout the year; P,; and P, are the amplitudes, @, and @, the phase angles of the annual and semi-annual terms. The time ¢ is counted from the beginning of the year, a month representing 30° in the annual term. R denotes the range, and > the sum of the 24 differences in the diurnal inequality. 24—2 184 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE TABLE XIII. Annual Variation. Declination (Unit 1’) Horizontal Force (Unit ly) He | 72 |) By Ie 2 wo 6, | P/M | P/M | P/P,| mM | P, A, | P, | 6, | P,/M | P/M | P.|P, (Max. | 9:55 | 4-10 | 277 | 1-40 7| 27 | 268 | -40 | -07 (Min. | 6:56 | 3-19 | 275 | 1-14 49 | -17 | -36 11-4 | 266 | 3-1 | 263 | -50 | -14 | | } (Max. | 49-4] 17-1] 277 | 48. | 304] -35 | -10 | -28 J239 |110 | 276 |17 | 243 | -46 | -07 |Min, | 30-6 | 14-7 | 273 | 4-2 | 291] -48 | -14 | -28°1132 | 72 | 265 |20 | 245 | -54 | -15 (Max. | 2°79 | 1-09 | 279 | 0-17 | 328 | -39 | -06 | -15 | 14-4] 7-7 | 275 | 1-2 | 254 | -54 | -08 1 (Min. | 1:63 | 0-91 | 274 | 0-15 | 311 | -56 | -09 7-9 | 5-3 | 264 | 1-2 | 231 | -67 | -16 (Max. | 2-19 | 1-00 | 273 | 0-28 | 280 | -45 | -13 | 28 | 7-8] 1:8] 290 | 0-6 | 293 | -23 | -07 2 (Min. | 1-56 | 0-78 | 275 | 0-20 | 267 | -50 | -13 | -25 | 4-4] 1-0] 265 | 1-0 | 249 | -92 | -22 (Max. | 1-05 | 0-31 | 280 | 0-38 | 283 | -30 | -36 | — | 31] 0:3 | 278 | 1-2 | 284 | -08 | 41 * (Min. | 0-81 | 0-29 | 273 | 0-27 | 275 | -36 | 34 | — | 2:3] 0-6] 249 | 0-9 | 299 | -27 | 37 | | Max. | 0-32} 0-15] 940-18] 279 | -46 | 57 | — | 14] 05] 104 | 0:6 | 301 | -33 | -45 ‘Min. | 0-28 | 0-03 | 71 | 0-15 | 268 | -11 | 55 | — | 1-2]. 0-1 | 278) 0-5 | 293 | -11 | -43 | | ° ° | 286 | -43 | -15-| -34 | 38-7 | 15-7 | 27 | 277 22-9 26 | | 28 In both D and H as we pass from sun-spot minimum to maximum P,/M and P./M diminish alike for the ranges, the sum of the differences, and ¢,. Thus, relatively considered, these elements are less variable throughout the year when sun-spots are many than when they are few. In H the semi-annual term is of reduced importance relatively to the annual term as sun-spots increase; but in D the dependence of P,/P, on sun-spot frequency seems small. In the case of c, and c, the absolute size of the element was so small that the values of P,/P, seemed too uncertain to record. Before discussing the phase angles in Table XIII as a whole it is convenient to consider an exceptional case presented by the annual term in ¢, for H. In the sun-spot maximum period the angle is 104°, which is comparatively near the value 119° for the 12-year period at Falmouth, and the corresponding value 126° for the 11-year period at Kew. But in the sun-spot minimum period the angle becomes 278°, or the phase is very nearly reversed. Suspecting some error, I calculated the corresponding angle for the sun-spot minimum period at Kew and obtained 290°, thus showing a close approach to the phenomena at Falmouth. In the case of a forced vibration, a change of this kind might arise from a change in the natural period either of the vibrator or of the disturbing force, that which was the shorter period of the two becoming the longer. Omitting c, in H, Table XIII shows in ten cases out of eleven a larger phase angle in the annual term in years of sun-spot maximum than in years of sun-spot minimum, and the same phenomenon is exhibited in the semi-annual term by all the six D quantities. This FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 185 means an earlier occurrence of maximum when sun-spots are numerous than when they are few. Asa check on the accuracy and generality of this result—corresponding data not having been given in (A)—I calculated the amplitudes and phase angles at Kew for the ranges, the sums of the differences, and c,, corresponding to those in Table XIII. In ten cases out of twelve—both exceptions occurring in H in the semi-annual term— there appeared as at Falmouth a larger phase angle in the maximum than in the minimum sun-spot period. §17. As the difference between phase angles is less suggestive than that in the times of occurrence of the maximum, I give the latter in Table XIV, giving only the means found in the case of Kew. Except in the case of the semi-annual term in H—where there are con- siderable differences between the ranges, the 24 differences and c,—corresponding Kew and Falmouth means approach closely to one another. The phase differences in D from the ranges, 24 differences and c, are, it may be added, in closer agreement at Kew than at Falmouth. Thus the acceleration of the epoch of maximum in the years of sun-spot maximum seems fairly established. The fact that an acceleration of the annual maximum should be combined with a retardation of the diurnal maximum seems a little curious. TABLE XIV. Annual Variation. Difference of times of occurrence of Maxima in years of Sun-spot Maximum (1892 to 1895) and Sun-spot Minimum (1899 to 1902), in days (+ denotes earlier occurrence in years of Sun-spot Maximum). Annual term Semi-annual term D H D H Ranges... un: a # se + 16 + 11:0 + 4-4 + 2-4 Sum of 24 Differences... an sie + 4:4 + 11-4 + 66 - 08 Gimmes. atts 3 shes as ate + 4:4 + 11:0 + 8:2 + 11:3 Cs — 15 + 25:6 + 6:7 = GH) C3 + 68 + 29-4 +42 - 39 Cha wane hee ae fos at re + 23:0 q + 5:7 + 4:1 Means from first three of above quantities | + 3°5 + 11-15 + 6-4 + 4:3 Corresponding means at Kew... ley | ee 34 + 10°3 + 7:0 — 31 186 APPLICATION OF WOLF’s FORMULA. §18. The nature of Wolfs formula Dr CHREE, A COMPARISON OF THE RESULTS FROM THE where S is sun-spot frequency, a and 6 constants, and R the value of a magnetic element, and the methods of determining a and b have been discussed in (A) and a subsequent paper*. Table XV gives the values of a, b and b/a in the case of the ranges and 24 differences for D and H for the several months of the year, also the arithmetic means of a and b from TABLE XV. (Units 1’ for D, and 1y for #). Values of Constants in Sun-spot Formula, calculated from Groups of Years Declination Horizontal Force Ranges Sum of 24 differences Ranges Sum of 24 differences a | dx104| (b/a)x104| a | ox 108 | (b/a)x10*| @ | bx 108] (b/a)x108| a | bx 102 | (b/a)x 108 January...| 3°19 265 83 13°76 | 240 | 174 12-1 161 133 AAT Sele 252 February. .| 3°32 427 129 15:63 | 351 | 224 88 261 296 53:8 | 133 248 Marchinss | 6:19 761 123 27:07 | 409 151 14:8 351 237 67:3 | 263 390 PANpyalleesr: 8-70 429 49 37°73 | 236 63 29-2 244 84 160-6 | 177 110 DMaiyiie-iere 8:25 495 60 35°71 | 316 | 89 26-2 317 121 165:8 | 214 129 June =e) 7:89 519 66 39:05 | 297 | 76 29-2 251 86 175-1 | 175 100 ail 7Bersaeee 8°32 457 55 38°97 | 259 66 28:2 261 93 1676 | 200 119 August....| 8-74 429 49 43°34 | 224 52 31:0 206 66 173°4 | 155 90 September 7:15 599 84 33°50 | 333 99 26-4 234 88 149-9 | 137 91 October an 6:24 355 57 27°47 | 269 98 24:2 154 64 146:0 | 104 72 November | 3°66 389 106 16°52 | 263 159 146 222 153 77°6 | 155 199 December | 2°25 277 123 10:75 | 211 196 5-9 165 279 34:0 90 264 | ee Mean of monthly values for | Weare sss 252 6:16 450 73 28:29 | 284 | 100 20:9 236 113 1 Su LS 135 Winter ....| 3°10 | 340 109 14:17 | 266 | 188 10:5 202 196 52°4 | 122 233 Equinox... 7:07 536 76 31-44 | 312 99 23°7 246 104 130°9 | 170 130 Summer...) 8:30 475 bili 39°27 | 274 70 28°7 259 90 170°5 | 186 109 * Phil. Trans. Roy . Soc., Vol. 203, A, p. 151. FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 187 the months forming the three seasons.and the year, with the seasonal and yearly values calculated for b/a from these arithmetic means. The results were obtained by grouping the years (cf. (A) § 52) and are immediately comparable with the Kew results in (A) Table XLI. It will be seen that b and b/a in both D and H tend to be larger at Falmouth than at Kew—z.e. sun-spot influence is greater at the former station than the latter—but the mode of variation throughout the year is remarkably similar at the two places. In the case of the H ranges the mean values of b and 6/a for the year at Falmouth are in excess of those at Kew by only 18 and 13 per cent. respectively, and yet taking individual months of the year the Falmouth value of } is the greater in 11 months and the Falmouth value of b/a in 10 months. Again in the case of the 24 differences in D the mean values of b and 6b/a at Falmouth exceed those at Kew by only 8 and 18 per cent. respectively, but there is only one month in the year in which the Kew values of 6 and b/a are the larger. This is far from what one would expect if chance were the dominant factor in individual months. The mean values of b/a for the year and the equinox are closely alike in all cases, as at Kew; also the winter value of b/a is notably the largest and the summer value the least. It will be seen that the value of b/a for the 24 differences is in excess of that for the ranges in every instance in D and in all but two instances in H. The ratios of the seasonal values of b/a for the 24 differences to those for the ranges vary in H only between 119 and 1:25; but in D the ratio is 1:23 in summer as against 1°72 in winter. The corresponding ratios for the year are 119 for H and 1°37 for D. §19. Table XVI gives the values obtained by the method of groups for a, b and b/a in the case of ¢,, ¢,, c; and ¢, for the mean diurnal inequalities for the year in D, H, N and W, and for the three seasons as well in D and H. The data correspond to those for Kew in (A) Table XLII, and closely resemble them. The sun-spot influence as measured by the size of b/a diminishes when we pass from the 24- to the 12-hour term, and from the 12- to the 8-hour term. The same tendency is generally visible in passing from the 8- to the 6-hour term, but to this the winter values appear an exception especially in D. § 20. The values of a and 6 assigned to the year in Table XV are arithmetic means from the monthly values. Table XVII gives values of a, b and b/a obtained by the method of groups from the mean diurnal inequality for the year; they answer to the Kew values in Table XLIII of (A). Taking b/a as the best measure of sun-spot influence, it is according to Table XVII fairly alike in D and W, and again in H and N; but it is considerably greater in the second pair of elements than in the first. The values of b/a for the 24 differences bear to the corresponding values for the ranges the following ratios: D W H N mean 1:24. 151 1-21 Ha 1:27 188 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE TABLE XVI- Values of Constants in Sun-spot Formula, calculated from groups of years. Fourier Coefficients in Expressions for Diurnal Inequality. Gh Os Cy Cy Element Season ee Se a bx 104 | (b/a) x 10% a b x 10+ | (b/a) x 104 a b x 104 | (b/a) x 104 a bx 104 | (b/a) x 104 Winter.| 0°69 165) 239 | 0-76 | 62 82 | 0-41 | 28 6s | 019 | 23 123 p | Equinox! 1:61) 179) 111 | 1-60 | 114 71 | 0-94 | 52 55. || 0-49) |» °¢ Dae Unit=1'| g mmer| 2:22 | 170| 76 | 201 | 102 | 51 | 089] 28 31 | 016 | —3) ae | Year...] 146) 174 119 | 1-46 | 91 62 | 0-76 | 35 45 | 0-26 19% 26 Winter.| 211 | 705 | 334 | 2:98 | 481 | 162 |1-08 | 140 | 129 | 0-85 | 111 130 y | Bauinox| 7°84 | 1036] 132 | 464 | 410 | 88 | 284 | 132 46...) 1-67 eae 16 Unit=1y| gummer|11-:14 | 1115 | 100 | 3:99 | 598 | 150 | 2-06 | 33 16 |0-71 |=36." eee Year ...| 6:89 | 922| 134 | 3-74 | 477 | 128 | 2-03 | 100 49 | 117 | 21 18 IN year 2| ye64 | 9904). 1308 481" | 530 | Tio) icy ot 7o «=| 081 | 13 17 Unit=ly Wi eaee lt zag) ecale teuemiey-26 | 407 59 «| 4-97 | 179 42 | 1-62 | 39 24 Unit=1ly TaBLE XVII. Values of Constants in Sun-spot Formula, from mean Diurnal Inequalities for the year (years grouped). Ranges 24 Differences Element a | bx 108 | (b/a) x 10° a b x 108 | (b/a) x 104 ID cece 5°72) 476 83 27:8 | 287 103 Te BeReico 20:2 | 2274 113 110:0 | 1509 137 Bir e0 20-6 | 2377 115 130°3 | 1673 128 FALMOUTH DECLINATION AND HORIZONTAL FORCE MAGNETOGRAPHS. 189 The corresponding ratios for Kew—obtainable from Table XLIII of (A)—are somewhat larger except in W, their mean being 1°31. The considerable excess in these ratios over unity shows that the difference between years of many and years of few sun spots may be materially under-estimated when attention is directed solely to the ranges, at least when these are based on hourly readings. Comparing Table XVII with Table XLIII of (A), we find for the mean values of the ratios of b/a at Falmouth to b/a at Kew, from ranges and 24 differences combined, D Ww H N 1-23 1:33 1:07 1:08 This points again to the sun-spot relation being more important at Falmouth than at Kew. § 21. In order to ascertain whether the connection between terrestrial magnetism and solar activity became closer when Wolfer’s sun-spot frequency was replaced by the area of faculae, umbrae or whole spots, I have calculated values for the a@ and b of (1)—applied to the ranges from the mean diurnal inequalities of D and H in individual years—making S represent in turn the Greenwich solar data described in § 3 and Wolfer’s frequencies. The method of least squares was used. From the values of a and 6b thus found ranges were then calculated for individual years. Tables XVIII and XIX show the excess of the observed ranges in D and H respectively over those thus calculated, with corresponding results at Kew for Wolfer’s frequencies. The tables also give the arithmetic mean of the differences irrespective of sign between observed and calculated values, and the “probable error” of a single calculated value. At the foot are shown the values of a and b in the formulae reached, and the value of B, where B is what b transforms into when the mean value of the variable S during the twelve years is taken as unity. The values obtained for @ and b in the case of Wolfer’s frequencies answer to those for Kew in (A) Table XLIV. It is immaterial to the fit of a formula of the type (1)—ze. to our estimate of the closeness what unit is actually employed. If, for instance, we multiplied all Wolfer’s frequencies by any constant n, the only result would be that every 6 would have to be divided by vn. The amount of agreement between the several formulae is most clearly seen by comparing the values of B. of the connection between magnetic ranges and the particular measure of solar activity It was certain @ priori for the reasons stated in § 3 that results obtained by the intermediary of the Greenwich projected and corrected areas would be closely similar. [ thus thought it unnecessary to take more than one series of projected areas, selecting the umbrae. The agreement between the results from the P and C umbrae will be seen to be remarkably close. The ranges calculated for the same year differ in no instance by more than 0°04 in D, or 0'°2y in H. In the case of H the formula from the corrected umbrae agrees slightly better with observation than that from the projected umbrae, but in the case of D they agree equally well. Weir, 00 MOS WIE 25 190 Dr CHREE, A COMPARISON OF THE RESULTS FROM THE TaBLE XVIII. Observed less calculated Ranges in mean Diurnal Inequality of Declination for the year. Greenwich Solar Data = = => : Wolfer’s Sun-spots Year Faculae Umbrae Sun-spots Corrected Projected Corrected | Corrected | Falmouth | Kew Teoiee ee PeesozB sl” 40-63 +063 | +050 +653 + 0-40 | 1992... .../ —069) | +042 | +038 | 4013 |) 72 0:i5 memos ieee eal) eats | +032 | +036 +020 | +034 | +089 1894.2, 0] 0-95 — 0:33 _o37 | -0-10 -012 | -0-16 1895 ... ut -0-41 | -0-30 — 0-27 -017 | -0-36 — 0-28 1896.) § Sal ese £016) |eeods +017 — 0-14 — 0-14 167. <2) 20:89 ~ 0:86 — 0:87 — 0-83 -051 ~ 0°53 HSCS see Uieie fs pO ~0-42 | —0-38 — 0:36 ~ 0-43 —0-41 1899... ...{ +012 | +0-24 + 0-25 + 0-25 + 0-15 + 0-07 TSOHen Mal arom) “4 Ogiellmclonis + 0-20 40-13 +0-01 19 Oe are + 0:16 + 0:05 | + 0:06 + 0:12 + 0°24 19020 eel 016 006 il) SOtonslesonG + 0-01 Reve Squseley } 0-457 | 0-332 0332 | 0-261 0-259 Probable Error _—...|_—:0°402 0-281 0-281 | 0-234 0-216 Valueofa ... ...| 6068 6-051 6-066 6-062 5-901 on Bove ee | 001215 | -01130 | -01559 | 002604 | -0451 | YOR BU PMes | MRED sr 1561 1565 1727 As was anticipated in § 3, the ranges calculated from the faculae differ considerably from those calculated from the other Greenwich data, especially near sun-spot maximum ; their accordance with observation is also conspicuously inferior. In the case of D the ranges calculated from Greenwich whole spot areas accord better with observation than those calculated from the umbrae, but not quite so well as the ranges ee FALMOUTH DECLINATION AND HORIZONTAL FORCE TABLE XIX. MAGNETOGRAPHS. 191 Observed less calculated Ranges in mean Diurnal Inequality of Horizontal Force for the year (unit =1y). Greenwich Solar Data = Wolfer’s Sun-spots Year Faculae Umbrae Sun-spots | | Corrected | Projected Corrected Corrected Falmouth Kew 1891... +34 +54 +5:3 +4:6 +4:8 +30 1892... ~4:8 +0°6 + 0-4 ~0:8 ~07 1-4 1893 ... +28 13 =12 ~1°8 =e +09 1894 ... + 8:1 | +. 1-2 +11 +2°6 +2°5 +0°8 1895 ... it ~0:7 -~0°6 0-0 -10 | —0-+4 1896 ... -12 +0-4 + 0-4 +0-4 Leper Sie 1897 ... } 4-9 ~40 — 40 ~ 38 — 2-2 ie 1898 ... =t =e ~16 —15 -18 -19 1899 ... | +03 +11 | +12 +11 +0°6 +03 1900 ... | -0-6 “iin | 20 —0°3 —0:7 +0°6 1901... ey | fetes +15 +17 42:3 1902 ... -25 -~1:9 1-9 29°] 155 ee \ 2-73 1-68 1-65 1725 | 1-70 Probable Error | 9-43 a7 | Maree 154 1-44 Value of a 21-11 20-77 | 2084 20-94 20-09 ee 00614 0589-0814 01339 2326 OS B 788 823 815 805 890 calculated from Wolfer’s frequencies. In the case of H there is little to choose between the results from umbrae, whole spot areas, and Wolfer’s frequencies; the last however still give the smallest probable error. 192 Dr CHREE, A COMPARISON OF THE RESULTS, eEvc. § 22. Excluding the faculae, the several calculated ranges at Falmouth for the same year agree on the whole decidedly better amongst themselves than they do with the observed ranges. The mean numerical difference, for instance, between the ranges calculated from the whole spot areas and from Wolfer’s frequencies is only 0°12 in D and 0°64y in H, the latter quantity being less than 40 per cent. of the mean difference between either set of calculated ranges and the observed ranges. Whenever the calculated ranges, excluding those from faculae, differ substantially from the observed—as in 1891, 1897 and 1898—all the calculated values are too large, or else all are too small. Further when the values calculated from Wolfer’s frequencies for Falmouth differ notably from observation, so also do those at Kew. There are two years when the calculated range for H is in excess at Falmouth whilst the observed is in excess at Kew, but in D the differences between observed and calculated values have invariably the same sign at the two stations. It would thus appear that the differences between the observed and calculated values, though comparatively small, cannot be ascribed in any large measure to accident, or to observational or local peculiarities. If the variation of magnetic range is due to direct solar action, then its intensity at a single station cannot be exactly expressed as a linear function of any one of the measures of solar disturbance considered above. § 23. In comparing the goodness of fit of Wolf's formula at different stations one has to take into account the extent of the variation of the element dealt with and the size of its mean value. During the 12 years considered the difference between the greatest and least of the mean annual ranges in D amounted to 3°95, the mean value being 7°63. The corresponding quantities in H were respectively 21-0y and 29°0y. Thus the mean differences in Tables XVIII and XIX between observed values and those calculated from Wolfer’s sun- spot frequencies represent 3:4 per cent. of the mean range in D, and 59 per cent. in H; whilst the probable error represents 5°5 per cent. of the excursion of the range in D, and 69 per cent. in H. The above agreement is about the same for D, but not quite so close for H, as the agreements found in a previous paper* at Kew, Pawlowsk and Katharinenburg. § 24. Looking at the values of a and B in Tables XVIII and XIX one cannot fail to notice the remarkable accordance between the results from the several Greenwich series of data. The agreement in the values of the constants derived from the faculae and sun-spot areas, for instance, is much closer than that between the constants derived from the spot areas and Wolfer’s frequencies, notwithstanding the much closer agreement between the ranges calculated from the two latter sets of data. * Phil. Trans. Roy. Soc., Vol. 203, A, p. 151. a, VIII. The influence of very strong electromagnetic fields on the spark spectra of (1) vanadium and (2) platinum and iridium. By J. E. Purvis, M.A., St John’s College, Cambridge. [Received 27 February 1906.] In developing Zeeman’s important discovery various experimenters* have measured the separation of many spectral lines of elements when vibrating in very strong electromagnetic fields, and they have shown, amongst other facts, that the lines of various elements which correspond to the laws of series are affected similarly, and that on the scale of vibration numbers the components of corresponding lines have the same distances in the same strength of field, and that the separations of the same type agree to the smallest detail for different elementst. The following paper is to describe the general effect upon a large number of the spectral lines of vanadium and of the stronger lines of platinum and iridium, and also to give the value of dd/d* for the constituents of the separated lines. The 21-foot concave grating spectroscope and the apparatus used in these experiments have already been described in a preliminary paper in the Proc. Camb. Phil. Soc., Vol. X11. Pt. 1. p. 82. It will be sufficient to state now that the strength of the field used in these experiments was 39,980 c.c.s. units. The maximum time of exposure was 30 minutes, and only those lines are noted which appear on the photographic plate after this period of exposure. The polarised condition of the constituents was determined by a calcite prism placed between the source of light and the quartz lens used to focus the spark on the slit of the spectroscope. The spectrum of the second order was used for wave lengths from about 3480 and extending into the ultra-violet, and that of the first order for lines whose wave lengths are below 23480. The column A gives the wave length of the undisturbed line; dd gives the distances of the constituents from the undisturbed line, a + sign meaning lines of greater wave length, and a — sign, lines of shorter wave length; dd/d? is multiplied by 10°; the letter s signifies that the line vibrates so that its electric vector is perpendicular to the lines of force, and p that it is parallel to the Imes of force. The remark “the middle constituent removed by calcite” means that the constituents of the divided line were so close, or so intense, or so broad, that they could only be seen separated when those vibrating parallel to the lines of force were blotted out by introducing a calcite prism between the source of light and the quartz condensing lens. The constituents of the divided lines are usually diffuse, and they vary in their relative diffuseness; some constituents are much more diffuse than others. This varying diffuseness renders the measurements very difficult; but for all lines whose constituents are more than three, there were eight or nine separate measurements; whilst for those lines which give triplets there were at least three separate measurements. * Preston: Royal Dublin Soc. Trans. Vol. vi. p. 385, 1898; and Vol. vu. p. 7, 1899. Phil. Mag. Vol. xty. p. 325, 1898; and Vol. xvi. p. 165, 1899. Royal Soc. Proc. Vol. uxm1. p. 26, 1898. Reese: Astrophys. Journ. Vol. xm. p. 120, 1900. Kent: Astrophys. Journ. Vol. xm. p. 289, 1901. + Runge and Paschen, Astrophys. Journ. Vol. xy. pp. 235, 333; and Vol. xvz. p. 123. Vor. XX ING. EEE : 26 194 VANADIUM. Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS The lines were identified by comparison with the numbers and descriptions of Exner and Haschek* for wave lengths above 4670°45. They describe over 2000 lines, and about 1000 of these are sufficiently strong to leave their images on the photographic plate in the time exposure of these observations. The two following lines are divided into si# constituents. The general appearance is that of two strong inside constituents which are more diffuse on the inner edges than on the outer edges, and two weak constituents, one on each of the outer sides of these. the two stronger inner constituents further subdivide. On analysis by the calcite d el |- +0610 | + 3:60 s | AX/? | +0214 | +1:25 s| +0175 | + 1:03 p 4123°70 Q@ | © | —O°175 | — 1:03 p | —0°214 | —1:25 s | — 0-636 | — 3°74 5 | The following lines are divided into five constituents ; centre of the middle one. +0°550 | +2:84 5 + 0°302 | + 1:56 p 44.00°80 0 0s! —0°308 | -1:59 p | -0°564 | - 2-91 s | +0845 | + 4:99 s | + 0:370 | +2:18 5 4116-70 ) O pit - 0-431 | —2°51 s ? Paes, The middle constituent is the strongest, and the two out- side ones are very weak. The middle constituent is ) much stronger than the others. The most refran- gible one, marked ?, widens into the constituents of the triplet of \ 4115-38, and could not be isolated. The least refrangible constitu- ent is very weak. The following lines are divided into fowr constituents. stituents of AX 443808 and 3692°38 is not very wide, and separation or if it is an ordinary reversal. | +0°617 | +3°12 s| +0224 +1°13 p| 444442 | (0) 0 — 0°224 | —113p | —0°614 | —3:13 s +0°468 | +2°37 s +0124 +062 p 4441-90 0 (0) —0:124 | — 0°62 p -—O'483 -2°44 5 |) he two middle constituents { are much sharper and stronger than the two out- | side ones. ae ae | * Sitz. d. k. Akad. Wien, 4438-08 | rd adn | dap | a easel AS chet | +0616. +3°65 s | +0211 | +1:25 8 } | +0°174 | + 1-03 p | 410998; O | oO I | —0°174 | — 1:03 p | | 0-211 | -1:25 s | | —0:626 -370s5 | and each constituent is measured from the | | +0:277 | +321 s There are three very narrow | ar +0" 141 +1:63 p | and sharp constituents, | 2034-48 | 10) O p and two broader and more \| —0:136 -—1:58p)| difiuse ones, one on each | ia 0°273 I Sep 17 Ss) side of these. | +0:266 | +3:16s +0126 +1 ae : | 2903-20; Oo | do. | -0135 | -1° ‘60 i | — 0269 | -3:19 s The separation of the two middle con- it is not easy to say if there is a real + 0°454 10) | —0:078 4436:34 | 1898, Vol. cvz. — 0-463 | + 0:078 | | + 0°533 | +0°118 | 0 —O-118 | —0°541 | | | The two middle constituents are much sharper and stronger than the two out- side ones. + 2°30 s | + 0°39 p do. IIa Abth. p. 184. ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 195 Newel edx dyn? d dy dX? { — — = - — - - — = | | +0657 | +3°33 8 +0421 | + = 08 s er) +0:130 | +0:66 p : + 07167 P| The inside constituents are 4421°82 0 ty) 3696-00 ) Py much stronger than the | —0°130 | — 0-66 p —0°167 | —1 en | outside ones. —0°633 | —3:24s —O0415 -3-05 s +0°820 +4205 +0°316 | +2°32 s ) +0°311 | +1°59 p| +0°093 +0°68 p |! There is no difference in the 4416°63 0 (0) 3692°38 (0) 0 - intensities of the four con- —0°311 | —1°59p —0°093 | — 0:68 p stituents. —0°831 | —4:26 s — 0°324 | —2°38 s +0°521 + 3°56 s +0135 | +149 p TONG) Ot eee (ee The constituents widen into 3013°12 0 0 those of 4407-89 and it was —~0°135 | — 1-49 p impossible to isolate them. ~0:328 | — 3°62 s | | Lines divided into four constituents by means of calcite. These lines have the general appearance of doublets, with the inner edge of each constituent sharp and strong, whilst the outer edges are weak and diffuse. On analysis by the calcite they are divided again, so that the lines are quadruplets with two inner strong and sharp constituents impinging upon two outer weaker and more diffuse constituents. 4210-02 | +0-323 4036°95 | 3977°88 | 3890°35 +0°174 0 | —O°174 | —0°323 + 0°284 + 0°166 0 - 0166 — 0:284 + 0:262 +0°178 0 —0°178 — 0-262 ance 10) nas ? s | + 1:66 p | 0 — 1°66 p © BS +1°909 s (+107 p| 0 | 1995 | —1:99 s 41°79 5 | +105 p 0 = 1°05 p | -179s +1°73 s + 117 p 10) =1:17p —173-s — 1:06 p The outer constituents are too diffuse and weak to Measure; only the two inside stronger members were measured. do. 3813°63 3298°89 3290°40 +1°71 8s) +114p) (0) — 1 -l4avp —17l1s | +146 s | + 1-04 p 0 —1:04p — 1°46 s | | +138 s +110 p | 0 lemeaee? | — 1°38 s | +1°34 s +127 p 0 = 127) 7p — 1345 as +1:08 p 0 —1:08 p eo. ee outer constituents are too diffuse and weak to measure. 26—2 196 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS x a | ap | N | ay ay)x? as ae Se -_ — : z = +0158 | +149 s | } | +0142 | +145 5 | +0:094 + 0°88 p | +0:110 | +113 p| 3249°71 0 0 3126°31 | (0) 0 | —0:094 —0°88 p | -0°110 | —1:13p —O158 -1:490 5 — 0°142 | — 1°45 5} +0°134 +1:29 5 +0°188 | + 193 5) +0103 +0:99 p | +0073 | +0-74p 3207°52 0 0) | euler) 0 0 —0°103 | —0:99 p —0:073 | —0°'74p}| | —0°1384 | —1:°29s5| . || | —0O°188 | —1°93 s +0°149 | +1:°45 5 | | ? Pees) | +0108 | + 1:05 p| +0:093 | + 0°98 p | The outer constituents are 3202°50 0) 0) || 3062-80 0) too weak and too diffuse to ities 07108 | — 1:05 p | —0°093 | —0°98 p measure. | - 07149 | -1-45 5 | > > |) | | ? DG | The outer constituents are | +0°312 | + 3°52 S)|\\Of thes tiwormndale stronger +0141 | +1°38p|| too weak and too diffuse to | +0°139 | +1°57p)| constituents the less re- 3193-29 0 | 0 _ measure. The less refran- 207570 0 0 - frangible one is slightly pee | F s | gible constituent of the two HI ss sell +120 “ne stronger than the more re- — 0:14 | = 1°38 p outside onesis strongerthan = ONSOn a ND. fran ‘ble one ip | ? s |) the more refrangible one, | —0°312 | — 3°52 ee ? | ? Penns ) The least refrangible consti- | +0128 | 41°45 +0°062 +0°62 p || tuent is stronger than the | 40-103 | + 1:23 Pp 3146-95 0 | 0 || most refrangible one. The | 2882-60 0 0 4 Aes |; outside constituents are too aires - - 0-062 | - 0°62 p | weak and too diffuse to | — 0103 | —1:23p ? Bes measure. | —0°128 | —1°54s +0143 | +1:46 5 + 07144 | +1°73 5) | +0°128 | +1:30p | +0°057 | + 0°68 p| 31350°40 (0) 0) | | 2880°4.0 0 0) — 0128 | — 1:30 p —0:057 | — 0°68 p - 07143 | — 1:46 5 | —0°144 | -1:73 5 Lines whose general appearance is that of doublets with the inner edge of each constituent sharp and strong, whilst the outer edges are weak and diffuse. They are probably quadruplets like the last but they are too weak to measure when analysed by the calcite in the time exposures of these experiments. The separations have been measured between the centres of the separated lines. = 0°261 __. | #07139 | +1:80 | __,, | FO101 | £002: |) FT i eee 4606°84 0) (0) | 3315:35 0) 0 and the closeness of the (OP IH2) |) TIeK0) | | —0-'101 | —0-92 J two lines makes the analy- sis very difficult. +0:189 | +1°30 | +ONIT | + 1:24 | 3807°69 0 0 3076712 0 (0) ~0:189 | —1:30 | | =O117 | =1:24 | | | | | +0:261 | +2714 | | + 07116 | +1:26 | 348001 | 0 0 | 3023-99) 0 0 =~ 2:14 | | —0:116 | —1:26 ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 197 N ar ON | RS dy|X2 oa | | | ‘ + 0084 [ee 0-93 ) The division is avery narrow | +0127 | +1°73 2996-05 0 10) . one, and the two constitu- 2708°00 0 | 0 —~ 0°084 | — 0:93 | ents are very diffuse. | —0°127 | —1-73 +0134 | +161 +0:093 | +1:26 2890-69 0) 0 2706°87 0 0) —0'134 —-1°61 | —0:093 | — 1:26 The following lines have almost the appearance of doublets; the difference is that in these cases the outside edges of the constituents are sharper and stronger than the inside edges, which are diffuse and blur into each other. On analysis by the calcite, they appear to be triplets, the two outer members of each being stronger and narrower than the inner one and it appears as if the latter impinged upon and overlapped the former. The distance between the two outside strong constituents has been measured because the middle constituent is weak and diffuse and it is as broad as the distance between the two outside constituents. It will be noticed that the constituents of these triplets do not vibrate in the normal directions, +0°265 +1°65p +0142 +1:40p ao0s12/ o | os 319809 0 0 s| | -0-265 | — 1°65 p| -0°142 | - 1:40 p +0220 | +151 p| +0°179 | +1:83 p| 3808-70 0 0s | 3133-48 0 0 s| | -0220 -1-51p —0:179 | — 1°83 p| The following lines are divided into triplets. The middle member of each is usually the strongest; and variations from this are noted. The constituents of weak lines are usually separated further apart than those of strong lines. The two outside constituents vibrate perpendicular, and the inner one parallel, to the lines of force, except where specially mentioned. The measurements were taken from the centre of the middle constituent. | + 0-402 | +1-68 4600°40| ......... lGeccebeeeee Too weak and diffuse to 4881°75 | 0 0 measure. —0°402 | —1:68 | | +0°322 | +1-52 | 4594-31 eed 0 +0°344 | +1°45 —0°328 | —1:°55 4875°66 0) 0 | —0°378 | —1°59 ) SS QUAT Ge se eet re Neto eis 8 do. | | | | +0°278 | +1°31 | | +0°312 | +1:48 4864-94 Oi)" 36 4586°55) 0 0 —0°270 | -—1:27 | | —0°308 | —1°46 | +0151 | +064 +0:292 | +1:39 4851°69 | 0 0 |The middle constituent re- 4580-60 | 0 0 0-151 =e moved by calcite. ~ 0-274 | — 1-30 | | STON Oe OS ae as Too weak and diffuse to | 4578°90) ......... Ih ose, || do. measure, | | +0:265 | +1-24 ) The middle constituent is +0190 +0:90 | J ; 4619-93 0 0) | Slightly nearer to the more 4577°36 0 0 The middle constituent re- | 0-265 | —1-94 { refrangibleconstituentthan _ ~0:190 —0-90 moved by calcite. 4 ) the less refrangible one. | | The middle constituent re- || | } | | moved by calcite, as the | +0°251 |) ar Ueto | middle and the more re- | 4572°00 (0) | (0) do. / frangible constituents were | | _0:251 _— 1-20 ) too close to measure, } a 198 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS 4564-80 4560°90 4553°22 4549°88 454560 4528°69 4524041 4489°11 4474-93 4469-92 4464-49 4462-60 44.60°52 44.5998 44.52°23 +0°310 (0) —0°310 + 0:296 — 0:274. | ane middle constituent re- } moved by calcite. do. Too weak and diffuse to measure. | The middle constituent re- } moved by calcite. Too weak and diffuse to measure. ) The middle constituent re- moved by calcite. Too weak and diffuse to measure. that it was not easy to | isolate the constituents, | and the measurement of the ( widens into 4459-98, so most refrangible constitu- ent is probably too high. Widens into 4460°52, and impossible to isolate its constituents. The most refrangible con- | stituent is stronger and more diffuse than the least refrangible. Themiddlecon- \ stituent also seems to shew signs of a division, but it is not very clear, and may be an ordinary reversal. | 4390-23 438488 4379°40 4353°10 4341-21 4353°05 4298-23 429787 | 4296°31 4292-01 4.284025 4271-68 426883 4235-47 | 4234-71 | 4.233°12 4232-66 | 4.232°20 ) | The middle constituent re- moved by calcite. Too weak and diffuse, and also the constituents widen into one another so that it was ) impossible to measurethem. The middle constituent re- moved by calcite. ——— These lines all widen and blur \ into each other; so that it / was impossible to measure the respective constituents. ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 199 4225-40 4205°30 4202°52 4183°67 4134-62 4132°15 4128°25 4115-38 4112°50 4105°38 4102°31 4100-00 4095-66 4092°86 4090°79 ay)? +1:12 0 — 1:04 | the middle constituent re- | moved by calcite. 4065°21 | 4053°76 4051-52 4051°13 4046°50 | 4039°76 4035°82 | 4023°53 4005°90 3999°40 | 3998-90 3997-28 3992-96 3990:72 _ 3973'80 3952°11 + 0254 — 0°254 +0:213 10) — 0:213 + 0:238 10) | — 0-238 + 0°2135 0 — 0:213 +0:255 | 0 | —0°255 | | +0°170 10) ~0:170 | + 0:256 | 0 — 0242 | + 0:257 — 0258 | | + 0-262 | 0 — 0-262 + 0°232 10) | —0°232 + 0:268 (0) — 0:268 | +0:200 0 — 0°200 + _ 2 lor | 2 | | | | | The middle constituent re- moyed by calcite. do. do. do. Too weak and diffuse to measure. || ame middle constituent re- | moved by calcite. Too weak to measure and widens into 3998°9. |) ane middle constituent re- moved by calcite. do. do. j do. 200 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS N N dy rN IN + 0°350 i t +0:244 | + 1°62 s9s420 | 0 tee mice comeiuaeAieo Mesemsi Oc ae — 0°350 ptt : — 0-236 | — 1°57 + 0:230 ‘) GRyAlSa | Eoncoa ||) coo das Too weak to measure. 3930-21 0 L do. — 0:230 + 0:308 | +2°05 8867°75 0 (0) + 0:236 —0°306 | — 2:04 3926°68 0 L do. — 0:236 +0°308 | +2°06 3866-90 0 0) + 0°192 =0'318 | —2:12 3926-45 0 + do. — 07192 ) , +0°203 | +1°35 The middle constituent re- 3865:02 (0) 0 ~ moved by calcite. 8925408 east Too weak to magerure and =10'203; | — 135 widens into 3924-86. + 0:214 an Lae +0°251 | +168 3924-86 0 ane eee constituent re" )B56:001. 1.0 o | do. ~ 0214 v ; _ 0-251 | —1-68 |} + 0:204 ) CSE Gocooe: \h-cacosk Too near 3856-00 to measure 3922-61 0 r do. exactly. — 0:294 } + 0°225 | +1-51 ; ; : The middle constituent re- RelA 3847°50 me a is ; ' moved by calcite. + 0:2 —0°225 | —1°5 3916°59 0 \ do. — 0208 } +0827 | +221 3844-60 0 0 r do. + 0:214 =0°327 | —2:21 391451 © 0) | do. — 0°214 | +0220 | + 1:49 3840-92 0 0 | do. S012 Salers eee Too weak to measure. 0-299 | — 1:49 } + 0°294 : L +0°184 | + 1°25 ) Ai 3 F The middle constituent re- ise 3910°05 0 f moved by calcite. 3828-72 | ) 0 elt do. — 0294 ) =(0-1840 |) 125 | + 0°308 ) + 0:294 | + 2:00 ) 3903-50 0 : do. 382713 0 0 . do. —0:308 — 0-294 | —2:00 |} + 0°284 ) ELT eco ods |se006 Widens into 3823°50 and 3902-70 0 do. difficult to measure. — 0°284: | ) | 882350) ...+- | eee eee Widens into 3823-37. + 0:240 | } PGPELOH | seacce || Soagoe : ae | Too weak and diffuse to 3899-32 0 } do. ie — 0°240 | WORCUBIMY soanoe || acocs + 0°344 +0°104 | +0°71 ) Th ‘dal tituent re- 3893-03 | 0 381810, 0 0 seoved by ‘datnitas Hamma — 0°308 —0°104 | —0-71 | + 0°230 ) +0213 | +146 |) Fi 387885 | 0 ao | 3815°50) 0 OF Bt es — 0230 f | —0°213 | —1:46 |) ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 201 d dy nx? d | ay adr | CUE RTT! | Seer (meri Too weak and diffuse to | +.0:277 | + 1-98 | Middle component removed measure. 3736'16 (0) (0) ~ by calcite as the three con- | +0:314 | +2°17 —0:277 —1:98 || Stituents overlapped. 3803°64 0 0 -—0°314 | —2°17 +0178 + 1°28 Sane _Middle component removed | a. 373298 0 0 by calcite. | +0:286 | + 1°98 | | —0°178 | —1:28 |) 3800:07 | 0 0 —0°282 -1°95 +0:256 | +183 | 73215 0 0 +0°312 +2°20 —0:256 | —1:83 3795-08 0 0 ~0°322 | - 9°23 +0120 +086 || : 3728°51 ) Ce iii 3 ROSEAON tan ccw|) Ate des Too weak and widens into —0:120 | —0-86 |) 3795:08. | | +0°318 | +2:21 +0°238 | +1-71 || | 3790-64 ) On | 3727°53| 0 OL; “a ~ 0322 2:24 ~ 0-238 | -1-71 |] + 0°302 + 2°12 5 7 SOOT SH) (ENERO a eae Widens into a weak line at 3787°39 0 0 | A and too weak to _ 0:97 90 e. o2i8 no +0318 +2-29 ates 8718°35 ) ) UM | Widen into each other and —0:302 | —2-18 Oa | difficult to measure. BUS OO riers. || Peco, a0 | vost! les 3715°70 0 0 Middle component removed +0°292 | +205 : O11 i by calcite. 3773°14 0 0 — 0-2 -—1°5 — 0:97 = Oe eet) | +0°262 | +2-00 | ee n-190:| “an | 3711-28 0 0 x +0°122 | + 0°85 Middle constituent removed ~0:292 —9-12 3771-13 0 0 by calcite. —0°19 — 0» VD) ES CUCISTEA (peta we Ia sean i Too weak and diffuse to measure. SOMES 4a ltrs See, || ecthe« Too weak to measure. +0:196 | +1-42 |)... +0°318 | 42:24 370490 0 o | oe ee removed 3760-40 ) 0) -0196 -1-42 |} —0°298 | —2:10 +0:252 | +1:83 |} +0°242 | +1-71 : : 3703-80 0 0 o: 3759-41 0 0 Shion removed 0-252 | -1-83 |} —0°242 | -1°71 rn = 701° Too weak to measure and cyte || ae Widens into 375010 and too 370118 Reema tad aster widens into 3700°5. weak to measure. +0°212 | +1°51 | 18 37 5 Middle constituent removed 700: ci ead ] Middle component removed 3750°10 0 OO Sees 3700°50| 0 0 ||. by calcite -0-212 | -1-51 |} > 0:188)) 0-37 ||!) cee { +0162 | 41:15 } . | +0406 +2:98 3746-00 ) Oa ht Ho. | 369043' 0 o || - 0-162 | -1:15 |} -0:394 — 2-90 l +0130 | +092 | + 0°322 | +2°36 |) an iti = 1 three constituents seem 3743°77 0 0 do. | 3688-21 (0) 0) to be equally strong. —0:130 | —0°92 | —0°318 —2°33 '] Vou. XX. No. VIII. 27 202 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS N a ann? | N an ann? ; +0°360 | +2:65 | SOSSIZSi lp <5) 2.50 eee Too weak and diffuse to 8683°25 0 : 0 | | ¥ measure, —0°356 | —2:62 | | +0°278 | +2°17 ; | | 357451| 0 oO} SlotsOhitey ||! Gamess || sooloce Too weak to measure. —0°274 | —214 | | } | S075:835| icky tll Secs | Too weak and diffuse to +0270 | +211 | measure, | 3573°21 0) 0 | +0°370 | +2:74 | — 0:266 | —2:10 | 367483 0 (a | — 0°368 | — 2°72 | | +0°201 | +1°58 | 3556-93 0 oO 4 + 0°320 | +2°37 | — 0:201 — 1°58 3673°50 0 0 | — 0°329 | — 2°43 +0:198 | +1°54 ) Middle constituent removed ; | | 3545°36) 0 0 | _ by calcite. SOT Saul wexcress | ll) teeeeders do. i} — 0198 — 1°54 oO As Too weak and diffuse to 3669°53 cs Bae ‘ ae | Middle constituent removed EN ae se 1) \ Sina measure. A ) by calcite. —0°203', — 1:50 + 0°326 | +2:60 | | 3538°88 | 0) (0) 3667°84 | ...... [Meas | Meieweaiwanidicdiiisa 46 | | —0°318 | —2:53 | | measure. | FNS |) coecse |, cmthernese ) | +0:216 | +1°72 3533-86 0 0 3661°5: +0°159 #118 ) Middle constituent removed ~ 0-208 - 1°66 3001°53 | 0 0 j by calcite. — 07159 = 118)" ) PLAS Ey MaGaueos IL asacdd 3646:02 | ...... | eevee | Too weak to measure. 3561635)! exenvoda sl keeeevaee + 0°514 + 2°38 ChrORO| Geasod || maacss \| 3627°83 | 0 0 (| vs) | —0°302 2°30 B522-02'|\ xtcocenlt sepeee Rs do. | +0298 | +2:26 | 3517 AGl tatss asc a OE 3625°71 | 0 0 | — 0°282 | = 2-14 SHURE) | coed |) ceacos | +0274 | +2-08 3507°69| ...-.- | sae. ) 3621°35 ) Os Ee | | —0:260 | —2°00 | +0109 +0°88 3504-58 On| are +0°184 + 1°40 ) Mi e —0:107 | —0°87 F , Middle constituent removed 3619-09 ) 0 If by calcite. —0'184 | — 1°40 | + 07184 + 1°50 | 3497°23 | 0 0 +0:220 +1°70 |) qs | -0:163 | — 1:32 3593-53 0 0 f a | —0:220 | -1-70 |) S40 3:07) seacoast. do. +0:194 | + 1°49 | +0182 ) +152 3592°19 10) (0) 3457 °30 0 (0) —0°190 — 1°47 —0°174 — 1°45 | +0157 | +121 |) S45 5051l) ae cual ieee | 3589°90 0 (0) do. : ~o157 | -1-21 j SAOA:CON er, aero | do. WuSBAGOS IR ysce eo Kas vt | ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 2038 3338-00 3321-72 3319-05 3318°04 331702 330462 3293-30 3288°47 3285°29 3282-69 3281-92 3281:26 3280°02 3277°88 3277°55 3276-25 327465 3271°27 | arr? d a | ane | | | +1:04 WiSRROL2S bev Sac hata siete : i 0 | The middle constituent re- | Sao Soe AE REE OD Lag iy } moved by calcite. } : I 5 “9 The middle constituent is 1°35 I 267-8: pe sale only very slightly, if at cf | 3207-84 Ons 0 all, stronger than the two ft) — 0°127 —1:18 |} ontside ones. — 1°33 +0°164 | +1°53 Se | 3266-06 0 0 ‘| | —0°159 | = 1°50 EET | : |\Too weak and diffuse to Daye 5 (oneal | ee +0145 | +1°36 elena |} 3258:°02 0 0 ‘| —~0°153 | — 1-44 ee | ‘ +0°125 | +1°17 EI sy | 3254 fi 0 || The middle constituent re- ee foe shi fey: | moved by calcite. i te e| a | +0197 | +1°86 trees | Too diffuse and weak to || 3252-01 10) 10) _ | Measure. —0°203 | —1°91 "4 Aas | } The middle constituent re- 40-158 41°50 aah moved by calcite. | 3250-90 0 0 =0°135 | —1°30 | +1°39 0 | +0148 +1°40 — 1°36 | 3238-08 0) (0) | -—0'148 |) — 1-40 re | 9 32 + 0°13 +1: ) : : ‘ banat _The middle constituent re- STN 3233-98 0 0 } moved by calcite. —0°139 | —1°32 + 1°88 0 +0°125 | +119 }) —1°81 323367 0 0 r do. ~0:125 | -1:19 |) + 1°45 0 +0:100 | +1:00 — 1°54 || 3232-10 0) 0 -—0116 | —1:10 +1°52 ) 0 j do. +0125 | +1:20 ) = 152 3227-05 0 ) do. —0:125 | —1-20 |) +131 | 0 do. +0:170 | +1:64 -1°31 | 3217°23 0 0 —0°159 | —1°53 +1:63 |) The middle constituent is 0 very little stronger than +0°240 | +2°32 = 1°60 the two outside ones. 3214-86 0 (9) — 0:224 | —2°16 na 9 Se Too weak and diffuse to | Eads | +0153 | +1-45 e three constituents seem “BF | } + 1:44 to be equally strong: the ie = ee oe a 0 middle one is only very Sel slightly, if at all, stronger than the two outer ones. 27—2 204 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS d a FONDS rd ay dy|X? +0:247 | + 2°39 ') The three constituents are +0°234 | + 2°33 |e peek (nee Simon ae 3208-46 | 0 0 |. very wide and diffuse into | 316491| 0 0 yea ee icant ou Be | Sot Sie on Use into : a Bir. | may account for the want — 0:228 — 2:9] e er. — 0:207 — 2-06 of symmetry. +0151 | +1:47 | +0:207 | +206 3205°70 Opsaty'li' 140 | 316813} 0 0 — 0-160 |) —1°55 | — 0218 —2:17 | | (Too weak and diffuse to CNOA) Sanades |) Koxeke | SUGERST | ooesec measure; and widens into | ) Too weak and diffuse to 3163713. SUGSCOG Tre -ce | ee rar eee: | +0217 | + 2:16 || 3161-42 0 0 | +0:203 | +1-99 —0:207 | — 2:07 3190-80 0 | 0 —0°191 | —1:87 *122 + 1:22 - i 3 | ND A 3158-01 oe A ) The middle constituent re- | +0164 | 41°61 “seul eee ) moved by calcite. 3188-60 0 0 —0°172 | —1-69 +0°139 | + 1°39 | 3155-51 0) 0 3187-78 a: Nes me The middle constituent re- SOON ie Ase J | moved by calcite. —0:110 | —1:08 +0:089 | +0°89 ] 3151-42 ) 0) do. +0°177 | +1°74 —0-089 | —0°89 3185-46 (0) (@) SOE |) = 17o SIA8*86)licvest een eee Too weak and diffuse to measure. +0°128 | +1:26 | +0°147 | +1-48 3184-04 0 0 || 3146-40 | 0 10) —07128 | —1:26 —0°143 | -— 1°44 | +0°136 | +134 S436 lal eecne sy a eeee | do. 3183°48 (0) (0) | —0°136 | -1-34 | +0°172 | +1-74 | 3142°67/ 0 ony +0168 | + 1:66 | -0°7159 | —1°61 SEH | OG | —O176 | —1-74 SUR EW Saodds I) aocose Too weak to measure and | widens into 3142-67, | +0-232 | +2-30 +0151 | +153 317417 0 0 314163) 0 0 — 0230 | — 2°30 I —0°160 — 1:62 | | | : | +0°201 | + 2:00 ot I +0118) +1°19 |) phe middle constituent xe 3168°21 | (0) | 0 : : || 3139-88 | 0) 0 |; moved by calcite. -0-201 | — 2-00 |} =O118H e119 | | | The least refrangible consti- | 0-191 + 1:93 | | tuent is wider and stronger | 313817 | 0 iy | than the most refrangible | | +0161 | +1:60 |} one, and it diffuses into | — 0184 | — 1-86 3167°55 0 (0) the inner one; it seems to | | —0:170 | —1:69 give some indications of +0°157 | +1°59 | “|| being further divided, but 3136-64 | 0 0 bi the separation is not de- ‘ OU finitely clear. | SORES SEW) +0-220 | +219 | +0164 | +166 316596 | 0 0 | 3135-08 0) OF —0:207 | —2:06 | | —0°164 | -1-66 | — Ea ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 205 d a =| aap } > an apr? | | | | Ee a | = +0191 | +1:95 I +0:096 | +099 |)... 3128'81 0 6° | PANOTOD oh MLS) 5Oics TP are eed To remoxeathe =0-191 | —1-95 i —0-:096 | -0-99 |) P | +0°191 | +2°00 +0°168 | + 1:67 3128-40 0 ) | 3094°33| 0 ) —0:209 | -2:18 | —0°154 | — 1-60 ; | + 07168 | +1°69 | i! Calcite used to remove the | 3093-23 0 0 ) all three constituents seem middle constituent. “q || to be equally strong. | 3125-52 0 0 —0:168 | — 1:69 —0°070 | —0°72 SLE eeeieoain | EBpeore | Too weak to measure and | +0°205 | +215 : | widens into 3125-52. 3086-61 0) 0 || Calcite used to remove the e middle constituent. IASAO) | cose | dese: Too weak and diffuse to | | -0-205 | —2:15 |) | measure. | +0°168 | +1:72 | +0166 | +1:75 3$123°01 0) 10) || 3083°31 10) 0) —0:166 | —1:70 | —0°164 | -— 1°73 +0172 | +1:76 +0213 | +2°24 3120°36 0 0 3082°65 0 —0:172 | -—1°76 —0°213 | —2°24 | AARNE ecieicar |P sere 2 do. OSTSO Ils octet |larerseta ete 3119°44 | | 308139 1! Too weak and diffuse to +0116 | +1:19 Caleite used to remove the | Orne) 2" soos ll caease Nearer 8118-51 0 oO L male constituent. All | three constituents seem — ic <6 = 0116 | -1-19 |} to be equally strong. — pA Dam ase } Calcite used to remove the | 3067°20 0 0 j middle constituent. +0139 | +143 -—07113 | —1:20 3116-90 0 (0) —0:139 | -—1:43 +0°186 1-97 | 3066-50 0 10) +0°130 | +1°32 | — 0-176 | — 1°87 3113719 10) 0 | —0:139 | —1°41 | SOO 5 Fill pesca lh erates Too weak and diffuse to measure, | |) Calcite used to remove the 311082 | 0 O4 |b do. || 3063-30| 0 OR | ors peeks eam —0:135 | — 1°39 ! —0°123 | ~ 1°31 i +0°149 + 1°54 } + O-174 +1°S5 | 3109°51 0 0 3060°60| 0 oF | -—0°147 | -—1°52 —0:164 | —1°75 | + 0:229 | +2-36 +0°143 | +1:53 | 3108°81 10) 0 : 3057 °55 0 0 — 0236 | —2-44 | —07149 | — 1-59 | NOG OOM | he veart-ceerey ler erie | Too weak and diffuse to +0174 | +1°86 | measure. 3054-00 (0) 0 + 0°202 | +2:09 —0O°1064 | — 1°74 3105:03 0 10) —0:205 | —212 | || +07135 | + 1°44 3053°48 0) 0 +0153 | +1°58 ') Calcite used to remove the —0-139 | —1:49 3102-8 0 0 |. middle constituent. All | three constituents seem c —0°153 | —1°58 | - SOSO;8 5s cclaee” rete are as Too weak and diffuse to ! to be equally strong. TeAgaTO. 206 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS N an ann? | » | @& | drape | | anes aee = | +0°160 | +1:72 | +0244 | +273 | 504900 0) 0 2989°67 | (0) 0 -—0°126 | —1:35 | —0°261 | —2-92 +0143 | +1°53 | +0:203 +2:27 3048-76 0) 0 2988:07 0) 0) —0°137 | -—1°47 | —0°205 | -—2-29 +0:097 | +092 | +0:131 | +1:47 |} ‘ || Calcite used to remove the | 2 | 3042°39 | aie athe i middle constituent. 2985°25 | ia ze | Des BOSH0S)| eo eo | wee |: get coe BR ainaiine do | +0172 | + 1-98 ae Me ene eee measure. 2983-62 0 | © |F one. The middle consti- +0159 | +1-73 ~0:172 | —1°93 |) tuent removed by calcite. 3033°99 (0) 0 | —0°168 | — 1-83 +0°283 | + 3:16 29082°82 0 0) +0°147 | +1:60 —0°275 | — 3°10 3033°55 | 0) (0) = 159 — 1°72 | iS “58 a a eS) Na | 2981-27 | ee as Re } The middle constituent re- . | | ae ‘| | moved by calcite. SUPBHO I sgecco Il) ses ana | do. —0'141 — 1°58 +0°337 | + 3°68 +0167 | +1°88 ) 302270 | O or 4 2976-55 0 o |r do. | —0°341 | — 3:73 —0°167 | —1:°88 |/ | | +0°219 | +2°40 | +O0:137 | +154 | 3016°81 0) (0) | 2074-06 0) 0 | — 0:27 | 9-38 —0°133 | —1°50 All three constituents are very broad and diffuse and +0151 + 1°70 HOM I cosdeo Il ako eas run into each other, and 2972°31 0) 0 | are not easy to measure | 0-143 | =—1°61 exactly. +0°234 | + 9°57 +0151 +1:71 3012-09 (0) 0 2968°40 (0) 0 —0:221 | — 2-43 —0°159 | —1-80 | Be oe Too weak and too diffuse to +0°298 | + 3:29 Mery week eae eC 100 measure: the three consti- 3007: i weak to measure, which men id a diffu 7:37 0 0 | may account for the want || o958-68| ...... | ...... BAe henner tan, UES —0:275 | —3-04 || of symmetr Zhe SORENSEN OIE © 6 into each other: the inner y cH | one is only a little stronger SOO Ta ee) meee than the two outside ones. -Too weak to measure. +0:167 | +1:90 ; : = y ; The three constituents are MOBAYA Ml SSo006 |) eassos 295754 0 0 very wide and diffuse. *| —0°167 | —1:90 | + 0'228 + 2°52 | 3003°50 0) 0 : . 2 k and too diff ti ~ 0-225 | —9-50 COO EHOW mocaod |) soooue peels 00 diffuse to +07180 | + 2-00 +0°186 | +2°13 | 3001°82 0 0 2952-12 0 0 | —0:182 | —2°01 —0:184 | —2:11 | +0:209 | +2:32 +0132 | +1°51 | 3001°28 0 (0) 2949-24 (0) 0) | -0°213 | — 2:36 -0:132 | -1°51 | —— ne ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 207 r dx ay|/? | r aX OND = | 2 +0137 | 41°57 | 2918-32 | ty Be Too weak and diffuse to 294815 0 (0) - | measure. 3 ~0°130 | —1°50 | -o-187 | + 219 ) The middle constituent re- : A ee Sie sist moved by calcite. The 2917-41 | 0 0 three constituents are very +0:203 | +2°35 —0°187 | —2°19 } wide and diffuse. 2944-68 10) 0 | cept eee 2916-00 ¥ A 65 ¥ Pat )The middle constituent re- Be 291 | moved by calcite. J 0G eee eee | 0165 | —1:94 |) ‘Too weak and too diffuse to z 2 ce la |r he peaenres SOLS AGN Ge sath Ih cae Fireteeemeeaete ee eo | measure. 2) | a ] 2914-97-| 22... | se eae +0°201 | +2°32 |) The least refrangible consti- | +0139 | +1:60 2941-51 0) | (9) tuent is more diffuse than 2914-40 | 8) 0 —0:199 | —2:30 |) the most refrangible one. | —o-141 | — 1°66 9 | AUSS S85) |) +1°88 2920°11 0 0 | 2893-47 0 (0) -0°166 | - 2-00 —0:146 | — 1°74 208 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS r an ayn|* r CN dX/d* oy I c +0094 | +115 d Babee a 20 |) + a eee used to remove the | 2849-19 | 0 0 | Caleite, used “A remoye the 2892 Werien tas middle constituent. | — 0-094 eae at middle constituent. | +0145 | +1°73 _| +0090 | +1°11 2892-51 0 0 j do. 2847°65 0 0 f do. —0°145 | -—1°73 l —0:090 | —1°11 +0:089 | +1:07 |) | _| +0188 | + 2-32 2891°78 0) 0) do. 2845°37 | 0) 0) ~0:089 | —1-07 |) —0°186 | —230 +0:132 | +1:58 +0°133 | +1°64 2888°36 0 0 | 2841-20 0 0 —0°149 | -—1-78 | —0:129 | — 1-60 g *B0"|, seam, IP aceree +0°11 + 1°48 ATCA ora \Neeat ! Weak and widen into each 2836-62 0 9 0 ) dor SRR TANG) | eee eeer | eee J other: too weak to measure. -o11g | — 1-48 | |/The two outer members are 9830°52l| 5 tee. eee. ‘ 172 eal | a little stronger and wider ) Too weak and too diffuse to +0172 | +207 P| than the middle one. The ||. ae | measure. 288491 Ox Oo s |, two outer constituents ap- BIB! soosc. || soonee | —0°166 | —2:00 p pear to vibrate parallel, and The least refranpable GaNeee | ho TES Une ae tuent is stronger than the fo Selle Snes os seh TOMS meet elaire most refrangible one and DSSOOQeweervce ales ees. Too weak and diffuse to 2810°39 (0) 0 { is almost as strong as the measure. | —0°124 | —1:57 |} ane one: nee ay rey 3 +0248 | 4 2-99 | Shee eee se : 6 : 2809'66 — 0°24 — 2°95 2809500) Norcal |b w= eters | hie weak and diffuse to O10 +121 |) Calcite used to remove the 2808°39| ...... | ----s- HRERSENAZ 2877°80 0 0 || middle constituent. : -~0'100 | —1-21 |! 2805:69)| eens eee Too weak and diffuse to -199 41°55 PAs lieite On Wwctoncio ce | osetia | ener Baastn + as 2 . + 07141 +1°71 | — 0°122 -— 1°55 2869-22 (0) 0 = (i) € — ities . 9 se 1°5 ee er ro) - Ons y Calcite used to remove the 7 a acta Be 28 ie middle constituent. ~ + 0-065 +0°79 | The middle constituent re- ea es > 2864-60 Or 0 ) moved by calcite. -0:065 | —0-79 | ee + ous & 0:97 ) a ( "OS r * +0095 | +116) n ~0-076 | -0-97 |) 2855°39 0 0 lf . ~0-095 | -1:16 |! +0127 | +1-62 |) 2798°88 0 0) if do. +0°118 | +1°44 —0°127 | —1:62 2854-41 0) 0 —O114 | —1:40 | +0114 | +145 |) | 2797-93 0 0) it do. PHOS || odsooe || bab ode | Too weak to measure. —O114 | —1-45 | | +0116 | +1-42 | . : +0°088 | +113 | 285136) 0 0 || Calcite removed middle con. 9797-12 0 o | do. -0116 | —1-42 |! i —0:088 | —1:13 | °° Ge aii ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 209 r dX dxr|/ Xr dn dX] QFBMEOR| oc -es || ces ees | +0:128 | 41°75 . 706: Calcite used to remove the } a oe and diffuse to || 2706-34 7 a 56 lhe ee middle constituent. SME herorersicss || eres revets +0°153 + 2:09 +0°149 | +1°93 |) Calcite used to remove the 2702°31 ) 0 2777°86 0 to) ' middle constituent. —0:151 | — 2:06 -0:149 | —1-93 |) +0°112 | +1°53 +0°111 +1°44 |) 2701°16 0 0 277481 10) (0) - do. —v0'104 — 1:42 -—O-111 | —1:44 |) +0:164 + 2:26 +0:098 +1:27 ) 2690°91 0 0 277440 0 10) r do. —0°153 —2°11 -~0-098 | —1-27 |) +0°159 + 2°20 +0°104 + 1°35 ) 2690°41 (0) 10) 2768°69 0 0 do. —0°168 — 2°32 -0°104 | —1°35 |) +0°172 | +2°37 +0°250 | +3:00 2689-99 0 (0) 2766°59 0 0 —0°159 — 2:20 —0:227 | — 2-98 +0168 | +2°32 +0147 | +192 2688-82} 0 0 2765°81 0 0 —0'161 — 2°22 —0°143 | — 1:86 +0°151 +2:10 +0:109 | +1:44 |) 2688-12 (0) 0 2760°62 0 0 j do. —0°159 — 2°20 —0°7109 | —1-44 +0°174 | +2°41 +0°185 | +2:43 |) 2683°21 0 0 Widen into each other, and 2760:26 0 (0) j do. —0°168 | —2°33 the least refrangible of one —0°185 | —2°43 \. almost coincides with the +0:168 | +2°33 most refrangible consti- +0-091 | + 1-20 }) 268298; 0 0 SHEE MEG GEE: 2753'°54 0) O j do. = 0°157 — 2:90 ~0-091 | —1-20 + 07157 | +2°20 +0135 | +1-79 } 2679°39 0 0 QT47-55 0 o | do. . —0:164 | — 2:29 0-135 | —1-79 |! +0°162 | +2:25 +0:081 + 1:08 ) 2678-66 (0) 0 2729°81 0 0 r do. -~0:159 | —2-21 —0:081 — 1°08 ) +0151 +2:10 +0:090 | +1°22 ) 2677°91 (0) 10) 271580 0 0 j do. —0:149 | — 2:07 -—0°090 | —1°22 F +0°137 | +1-91 Pee. é 267211) 0 0 ALCS ro nos |||" ocoene poe ee too diffuse to 0135 | —1-90 2711-88 : ae ¥ ap 2663:42 i is, oe lhe ree ie middle constituent re- _0-188 | — 2:55 0-094 | — 1°32 moved by calcite. Von, XX: No... VEL. 28 210 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS rd an adr? nN a | an 6 s eae | Constituents overlap and are BOSS BAO wioneers IIL) sarees Mipheyeae eee ALES | 2528°59| ...--- hese ce too broad and diffuse to f measure. [ML SEGRE QOA5:O0)|| teeta cea Nl) wietersiele ) | +0°115 | + 1°80 | 252800 0 | oO + 0°180 + 1:86 —Or115 —1°80 | 264.2°32 0 0 ) Very weak, middle constitu- | sale aan cose } ent removed by calcite. 2524-07 | \ | | | +0108 | + 1°56 | 2521-62) ...... | ae eke | 2630°72 0 onl aed | | —0°108 | —1:56 | 25116:19))) eer |) SG araie +0120 | +1°78 9476-60) urtha.nt.. 1 ce < cate 259520 0 0 | ae. | { oo and diffuse to —0:120 | —1:78 || 2479°09| ...... leaner | | | CANOE RCT | os coeoc || eomood | 2405°30)) i224... Weed. cos Too weak and diffuse to | | | BboS MM | yee et If weer eo eee \| 2808*70)! a5. 40: | eee | | | +0:123 | +1:89 O899-FOl lhe aes a sesncten: ) 2549°36 0 0 —~0'19 = iiie | . | +42 A pile ase 2371°10 | e a We a || The middle constituent re- 237119 | | moved by calcite. + 0°087 + 1°34 = 0-080 | — 1:42 2528-97 0 0 | | -0°087 | - 1°34 || 2366-40] ...... Mpavese Too weak and diffuse to | | measure. The following lines became doublets, and most of them are very diffuse and weak; and it is probable that other constituents overlap. In each case the width between the centres of the aggregate of the two constituents has been measured. | le | | +0°164 | +1:08 +.0°179 | +144 | 3885°03 0 | 0 | 3520719 0) | 0 | — 07164 | —1:08 -0179 | —1'44 | | | +0:203 | +1:35 +0°172 | +1:41 | 3876°25 0 | 0) : 3486-09 (0) 0) —0°:203 | -— 1°35 —O-172 | —1-41 | | | +0153 | +1:02 | +0°109 | 41:18 | 3864-00 0 0 8043°62| 0 0 —0:153 | — 1:02 |} -07109 -—1'18 | +0180 | +1:41 | +0:120 | +1:29 3566°32 yy Oo) || 3041-52 Ooo) —0°7180 | -—1°41 | —0°120 | —1:2 +0148 | +1516 |), y Mal | | 356078 (0) 0 \ ia 5 Me ee are || /A very narrow separation: —0:148 —~1:16 } BESOuES i | the constituents are very | 3008-61 diffuse and the more re- Sane e es te Noe fe aa | costs || frangible one is stronger +0°387 | +311 | and sharper than the less 352489 Om 0). is | | | refrangible one. —0°387 | —3-11 ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 211 The following lines do not seem to be either divided or widened; they appear to become a little sharper and weaker when vibrating in the magnetic field :— 4330°28 2910°50 396819 2889°71 3896°32 2775°89 2950°40 2739°80 There are numerous other lines which are widened, but they are too weak to examine or separate by the calcite; and others show a weak diffuse widened middle constituent, whilst the two outer constituents are too weak to affect the plate in the maximum time exposure of these experi- ments; they would probably become visible if an exposure of two or three hours were given. PLATINUM. The spark spectra of platinum and iridium have a large number of lines, most of which are weak, Only the strong lines are noted and whose images were photographed during a maximum exposure of 30 minutes when vibrating in the magnetic field. .The lines were identified and compared with the descriptions of Exner and Haschek*. The following lines are divided into triplets with the exception of those specially described in the notes. d aX apr? rd dy ay|X is (abs triplet could be measured iis ; / +0289 | +1°66 || only by removing the con- +0°186 | +1°70 4164-9 0 (0) | | stituents vibrating parallel 3301°9 (0) 0 _ 9-926 = Fie to the lines of force by =a iiss This has the appearance of 0°289 1-66 { means of the calcite prism. | o180 1°65 a doublet not unlike that | } a of \ 3408-30 with the inner +0286 | +1:68 |) +0°146 | +1°36 edges sharp and the outer 4118°9 (0) 0 } do. 32740 (0) 0 '\ edges weak and diffuse. It _— 0-2 _ fe | —~0:146 | —1°36 || is probably a quadruplet 0-286 1-68 | | like 3408-30, but it is too k to determi - 40-294 | 41°86 r 40159 | 41:50 ae o determine accu 3966-4 ) On| do. 3256:0 (PN oO —0°294 | -1°86 |! —0166 | —1°56 | | +0°286 | +1°81 ) +0157 | +1748 . . < : The middle constituent re- 3925°6 ae Bie | do. 3247-6 ates ae moyed by calcite. — 0-2 = px —0- 7 — ]* | +0°320 | +2-19 |) i +0°192 | +1°87 3818-9 0 (0) r do. || 3204-2 0 0 —0°320 | —2:19 ) HT —0°195 — 1°89 +0:262 | +1-98 + 07199 | +1:94 36283 | 0 0 do, | 3201-0 0 0 — 0-262 | —1-98 | —0°195 | —1°90 The general appearance is } This has the general appear- that of a coon with ae +0123 | +1:24 ance of a doublet like that +ougy |siae|| ee, Sete. am | sisos | "0° | “ot } Sci se og +0°122 | +1:05p|| weak and diffuse. On —0:123 | —1:24 and diffuse to measure. It 3408°5 0 0 analysis by the calcite, it is probably a quadruplet. = ort ae eee eee een +0178 | +1°89 ~ 0167 | — 1-445 stituents impinging upon | 30648 e a and overlapping the two | \ 0-178 | —1°89 outer weaker ones. * Sitz. d. k. Akad. Wien, 1896, Vol. cv. Ia Abth. pp. 523, 542. 28—2 212 Mr PURVIS, ON THE INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS d wm | aye heel Poaran arr? | +0°105 | +117 E . +0112 | +1°45 |) The middle constituent was 3001°3 0 0 | The same remark applies to || 2771-6 0 0 | removed by the calcite ~0-105 | -1-17 |) : —0:112 | —1:45 prism. +0178 | +2:00 +0:130 | 41:73 | 2998-0 OG. yi 30 2733'9 OF in —0°184 | — 2:04 —0°128 | -1-71 | +0157 | +1°82 | 40149 | +201 2929'9 0 0 2719'1 0 0 —0°163 | — 1:89 —0°159 | —2:15 | +0°186 | + 2:22 | +0147 | + 2:00 2893°4 0 0 27059 | 0 0 —0'189 | — 2°25 ~0:145 | — 1-98 | | +0154 | + 1°92 | +0°141 | +1:93 2830°3 10) 0 2702°5 0 (0) | —07149 | -—1:86 —07140 | -1°91 | +0135 | +1:73 +0:091 | +067 | 2794-2 0 0) 2659°5 | 0 0 | do. —0°132 | —1:70 | —0:091 | -0°67 | | +0°143 | +1°85 |) The middle constituent was | 27748 (0) 10) removed by the calcite | —0°143 | -1°85 prism. | In addition to the lines described above there are other lines which are divided into triplets, but the constituents are too weak to measure with any degree of accuracy, Such lines are the following :— 4552°6 4498°9 44.42 °7 4192°7 3672-2 36433 3639°0 35532 3485°3 2 3252-0 2890°5 27550 2677°3 2628'1 ? 2425-0 23773 2310°9 2 And there are a number of weak lines which are widened, but they are too weak to shew any separation when analysed by the calcite prism, ON THE SPARK SPECTRA OF (1) VANADIUM, AND (2) PLATINUM AND IRIDIUM. 213 IRIDIUM. ae ae | aap? | rd a | arr | This has the general appear- | | |{ ance of a doublet with the +0192 | +2:03 | +0°298 | +2:06s|| inner seen sharp and || 3069-0 Os ih. 40 107 | : strong and the outer ones nS _ 2-00 cone | sabe | ae 36 p | weak and diffuse. On aes) - 3800°2 0 0 analysis by the calcite it | ‘ -—0197 -1:36p separates into a quadruplet, | This has a general appearance —0°298 | -2-°06 s with the two inner con- +0118 | +1:27 similar to that of 3800-2: stituents impinging upon || 3049-7 0 0 | but the two outer consti- the two outer ones. ~ || tuents are too weak to af -07118 | -—1:27 . 1 “ye analyse. It is probably a | +0:266 + 2°04 quadruplet. 3605-9 0 0 HI | -—0°254 -—2:°00 |} +0°150 | +1:73 | 29433 | Oo | oO +0°316 | +2°47 | | -O141 | 1°62 3573°8 0 | 0 } | —0°306 | - 2:40 | | +0184 | +2°16 || 2924-9 0 0) +0314 | +253 | -0-184 | —2:16 3516°0 Oro, 0" } | -—0°310 | —2°50 | +0°157 Heat 1:93 | | 284977 | 0 0 +0°192 | +1°85 | | — 0°154 — 1:90 32207 | 0 ay, | | | -O-189 | -1-82 | | £0174 | +216 || 2883°3 | 0 (0) +0194 | +1:97 -—0:167 | —2:08 3133°4 Ci se | | | -O-192 | —1-:95 | | i \ | \ In addition to the above lines there are others divided into triplets, but they are too weak to measure ;° such are :— 40703 35945 2774-9 536532 3573'8 26646 5628'S 2824'6 2368'1 There are also others which are widened, but they are too weak for analysis by the calcite. As a rule in both platinum and iridium the middle constituent of each triplet is the strongest, but the constituents of the triplet of Pt 30648 seem to be equally intense. Also the constituents of the normally weak lines are always separated further from each other than those of the strong lines. 214 Mr PURVIS, INFLUENCE OF VERY STRONG ELECTROMAGNETIC FIELDS, ete, GENERAL SUMMARY. The more important conclusions which may be drawn from the preceding observations and measurements are :— (1) The values of dd/d* for many lines are essentially identical. The general appearance of the corresponding constituents, and that of the normal lines, are also in these cases very similar. The most striking illustrations of similarity in the case of the vanadium lines are the sextuplets derived from 4123-7 and 4109-98, the quintuplets from 2934°48 and 2903°20, and the quadruplets from 444442 and 369600; and amongst the lines which are divided into four on analysis by the calcite there are several which are very similar in their appearance and in the values of dd/d*, Also amongst the lines becoming triplets there are lines which can be grouped together in a similar way. The method, therefore, may be a means of correlating and classifying into groups having the same general properties the spectral lines of elements which have not been classified in the same manner as those of the alkalis and the alkaline earths. (2) The metals platinum and iridium are usually grouped together in the same family having the same general physical and chemical properties, and the method may be a means of correlating and classifying corresponding lines of elements belonging to the same family. Only the strongest lines of these two elements were photographed, but amongst them it will be noticed that there are several lines which have the same essential value of dd/d? for the constituents, and the general appearances both of the constituents and the normal lines are in these cases essentially the same. (3) In some instances the values of dd/A* for the several constituents of the same line seem to be simple multiples of each other; and also the value of da/A* for the constituents of some lines is a multiple of the value for the constituents of other lines. The results of the measurements of other metals will be published later. Finally I desire to convey my thanks to Professor Liveing for the use of the spectro- scope, and to Professor Larmor for his sympathetic interest in the investigation. IX. On the Asymptotic Expansion of the Integral Functions 2 n x ah 5 a” (1+ an) ip Gee (1 +780) azo F(x) n-o /(l+n+n6) By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge. [Received and Read March 12, 1906.] § 1. IN a memoir “On the Asymptotic Expansion of Integral Functions defined by Taylor's Series*,” the author has discussed the expansions of various wide classes of integral functions. The results obtained in the present paper were indicated, though no detailed investigation was given. Such an investigation is now supplied. The functions are of interest in that the associated functions have not finite radius of convergence. They may well be considered together as their asymptotic expansions are related. And the method employed to obtain their asymptotic expansions in a region including the positive half of the real axis differs from that which was used for any of the other functions considered, depending, as it does, on the use of Lagrange’s series. : ee eee LG! The function represented by the series = * +9") ve denote by f(z, a); that aR Gea) 2 aT (1+76) . tis eke: J y = : represented by 2 TG n ind) by (a, @). We shall assume that 0 0. When a=1, f(z, a)=5~ 3 when «=0, f(x, a)=e*. Both the functions f(z, a) and < T'(1 + an) n=o (1+ Bn) which also includes as a particular case Mittag-Leftler’s function H#,(z). This function the author hopes to consider on a future occasion. 1+¢(z, @) are particular cases of the more general function xz”, (8 >a>0), § 2. The asymptotic expansion of f(x, a) when R(x) <0. If an be real and >—1, we know that T (an+1) = e vy" dy, 0 the real positive value of y™” being taken, and the integration being along the positive half of the real axis. * Presented to the Royal Society in May 1905 and shortly to be published. Wor. XX. No: EX. 29 216 Mr BARNES, ON THE ASYMPTOTIC EXPANSION Hence, if 0 ——- ent e(n+i)/a—1 dt. oC JO n=9 Uv" Now I have shewn in my original memoir (§ 5) that if we replace the previous integral by the sum of the integrals of the various terms in the subject of integration we obtain a series which is truly asymptotic. We therefore have : 1 X © Pr{(n+l1)/e T= x, a)=——, = ( T : 1)/ ax? j25 T(mt+l1)2 where |Jya'¥*«| can be made as small as we please by making «| sufficiently large. N> Finally therefore, if R(w)<0, and 0| arg a|> (1 —a) 7/2. We now proceed to shew that the previous expansion is valid when 7/2 >| arg 2 }>(1 —a) 7/2. Suppose that 6, the argument of «, is positive and equal to 7(1—7)/2, where 0<7/2. Hence the subject of integration tends exponentially to zero along an are of the great circle at infinity for which the argument lies between 0 and —¢, the limits. included. The integral can therefore be taken along the real axis, and we have 2 x nN pmudb 2 (—z)"e KE QVESE i exp {ae z*} > ier Det dz n=0 n ! ee xn 2 —)" (n+1)/a—1 emed Ep =—> eu( WF J ad\mtva CY asymptotically. a n=0 /0 nm! (— xe”) And thus we have the asymptotic expansion a (Tr (" ‘i =) SF (@, a) =— > SSS @ n=9 (n+ 1)(— ays We may obviously employ a similar method when 6, the argument of a, is negative and w/25| 0|>(1 —a) 7/2. § 4. The asymptotic expansion for f(,a) which has been obtained in §§ 2 and 3 is thus valid when | arg(—)| < (1+4) 7/2. We proceed to verify this result by means of the contour integrals employed in the fundamental memoir. Let us consider the contour integral - = {ian (—s) Tl (as+1)(- 2) ds, where the contour of integration embraces the positive half of the real axis and encloses the poles of T'(—s), but none of those of ['(as+1). Its value, by Cauchy’s theory of residues, is evidently f(a, 2). Now when |arg(—2)| <(1+a)7/2, the integral will vanish when taken round that part of an infinite contour for which R(s) is greater than a finite negative quantity however large. This theorem is immediately evident from the known asymptotic expansion for I'(s), when |s| is very large. The integral may therefore be taken along a contour in the finite part of the plane which consists of a straight line parallel to the imaginary axis which passes between the poles —k/a and —(k+1)/a, together with a loop which encloses the poles —1/a,...,—k/a of T (as + 1). We therefore have, if |arg (— #)|<(1 + a) 7/2, f(@, a= = Pin/a) (2)e* at el (—n+1+ae)+ Jz (a) k (= VvEn = 3 eh td (z) where | J; (x) z**| can be made as small as we please by taking || sufficiently large. This is equivalent to our former result. 29—2 218 Mr BARNES, ON THE ASYMPTOTIC EXPANSION § 5. The asymptotic eapansion of f(a, a) when | argx| <7 (1—)/2. We will now shew that, when |arg.x|<(1—a) 7/2, we have the asymptotic expansion St (@, a) = exp {1 = a) qa/(i—a) gil Ge)} (ax) (1—a) [ 2 T (n+1/2) a {* —aT —(T+ ee | S ; n=o ' (2n+1) (an)Ve-2) Qp aT? T=0 We have seen that P J (a, a) = [exp {—y +ay*} dy, the integral being taken along the real axis. Suppose now that we make the substitution = (ax)¥a-9) ie Then the integral becomes i exp [(aax)/0-«) {t/a —s t}] (aa)! 0—a) dt, 0 taken along an axis such that arg¢=-— {arg a}/(1—a). If now |arga|/(1—a)<~7/2, this integral may be taken along the positive half of the real axis, for the difference between the two integrals will be an integral along an are of the great circle at infinity which vanishes. If then we put X =(ax)"""%, we have, if | arg «|< (1 —a)/2, F(a, a) =X [ exp {Xi*/a — Xt} dt, the integral being taken along the real axis. Make now the substitution y=(1—-t)/at+t—1, all the quantities involved being real. Since a<1, we see that, when ¢ is very large, y=t. When t=0, y=1/a—1. When =1, y=0, and we have the minimum value of y. The curve which represents the connection between y and ¢ will be given by the figure. t=] The line y=1/a—1, cuts the curve again when t=q¥@, We obtain upon substitution ;

0 we must take the positive value of y'*, and when 7’ <0, the negative value. where = | Evidently d, =V2/(1 — 2). To find the actual radius of convergence of the series (1), we must adopt the process of Nekrassoff* or some analogous method. — a) —1/2 pee ote , one or other of the two values of the root being Let vie= 5 definitely taken. Let 7 be a region of the z plane containing the origin within which the function z¥(z) is regular. Draw within 7 a contour S containing the origin but excluding the points for which z is real and <—1: such points lie on the cross-cut which serves to make (1 +z)* one-valued and for which w(z) is not defined. Then, as Cauchy first shewed, the equation pr an[l tal (1+ Va will have one root JZ’, which can be expanded in terms of Vy by the series (1), within the contour S, provided on the contour S, y'*W(z)\< 1. We must find that contour S on which |y(z)| has the smallest possible value if we are to obtain the largest value of |¥*| for which the series (1) is convergent. At each point of the domain 7 erect a perpendicular to the plane of length M= y(z)|. We thus get a surface which is cut by a plane M=M, in a curve. When W&M, is very large this curve consists of a series of ovals round z=0 and the other zeros of 1+az—(1+2z):. As M decreases these ovals expand until finally the oval round z=0 touches some other curve, let us say, when M=k, * Nekrassoff, Mathematische Annalen, Vol. xxx1. pp. further references see Osgood, Encyklopddie der Mathema- 337—358. Cauchy’s original development will be found in tischen Wissenschaften, Bd u. 2, pp. 44—47. the Ezercices d’Analyse, etc., 1840, T. 11. pp. 41—98. For 220 Mr BARNES, ON THE ASYMPTOTIC EXPANSION When M | ey X yo? 2n+1 : n=0 - yd 0 Gaye =|"¢ wore o - = | eo x y 2n+1 2 On)! ! y" dy N-1 72 if 1/2 fa > g(b—Y) X giz —2N+1 an eae ey) dy an i y (Qn) 14 dy eX = = n+N—1/2 Therefore | Jvl< Peg call G arg X)e S donzons1 ¥ 0 n=o A" (20 ayy | rh — e +|eer |e / e~ (y +p)? Qp+ yay ld. tle xf Neb a (y +p)" 1/2 ay|, Jo n=0 (2n)! the first integral being taken along an axis whose argument is equal to the argument of X which is < 7/2. dt 1 § 8. Now dae i it behaves like —y'* Hence |JyX| can be made as small as we please by taking |X | sufficiently large and jarg X |<7/2. The theorem is therefore established =? and this is finite on the range 1/a—1 to «, and at infinity ao Hence eto | er a lja-1 «| dy, where e>0, tends to zero as |X| tends to infinity provided |arg X|< 7/2. Finally therefore if |arg2|<(1—a) 7/2, we have the asymptotic expansion dont D(n + 1/2) ee nao RQ) ix 42 a where |Jy| can be made as small as we please by taking |X| sufficiently large, and X= (axle), Cola Xexp(X(ja—1j {= ; 222 Mr BARNES, ON THE ASYMPTOTIC EXPANSION This result may be written F(@ a) = exp {1 e a) q2/ (1a) apie) (aa)! (1—a) ¢ ee r (n + 1/2) qd” i} +aT = (74-1) ties nao (Qn 1)(ax)v°—9) aT™ aT? m=9 |. The first term of the series is {2ar/(1 — a@)}1*. § 9. An interesting deduction from the previous formula may be noticed. The author has for a long time speculated as to whether it was possible to con- struct an elementary integral function which should tend to zero for all values of || round infinity except those within a range of values of arg as small as we please. being, of course, infinite within the latter range. Such a function is given by aT (1+n—n/p) =O (1 +n) where p is a large positive number. o ea Sy For if we put a=1—1/p in the previous formula, we see that, within the range jargx|<7/2p, the function admits an asymptotic expansion whose dominant term is i / p—\yrr_ exp re (1—1/p)? a? l (et ==) (Qamp yi aelsesoscecnsececeeeeenne (A). Outside this range the function admits an asymptotic expansion whose dominant term is ifs >) it ) ( »\1+1/(p—1) B Has owess (HYD) waned nse cones: «ele “* — pee ey IES 70) § 10. In the preceding general formula we have only obtained the dominant exponen- tial term in the region argz|<(1—a)7/2, and for all we know to the contrary other exponential terms of lower orders may exist. In the special case when a=1/2, the corresponding series of Lagrange is finite, and we can shew that no such terms exist: we can also bring out more clearly the dis- continuity in the asymptotic value of the function, ea bo bo oo OF CERTAIN INTEGRAL FUNCTIONS. 2 oT (14+n/2) AN a2 BS Let f@= >= Tan) rl Then, as before, f(z)= | exp {—z+ «2"*}dz, the integral being taken along the real axis 0 from 0 to «, and the positive value of 2° being taken. The previous transformations in the special case now considered lead to FES TSE UR Be ee sop C TORE OOCE CORE SCOR OC ORO RDORCECCUEICOOE (1), dz 2 =e |. Yen ee 2 é —— a whence Z+22 y ta 4 and ts ope We thus have Fee eee z | a) = | e**-Y 51 + —_} dy. a J aber aes The transformation (1) is the usual parabolic transformation. If 2/2 =p+.uq and y=&+u, we have, corresponding to real values of z, 1° = 49° (E + q°). This is the parabola DACBE, with its focus at the origin, of the figure. We will consider three cases D (a) when 01, the series is divergent but summable. We may therefore substitute the series under the sign of integration to obtain the asymptotic value of the integral. Therefore, when |arga|<7/4, we have the asymptotic formula X (—)" T (2n + 2) een T(n+1)a™ where | Jya?7| tends to zero as «| tends to infinity. = 2 (2) = lara ces = F(x) = wae z 12 § 12. Consider neat cases (8) or (ry), when m/2 (x, O)= ey Paes ne) § 13. In the series by which we define the function ¢(#,@) we suppose that @ is real and >0. Evidently when @=0 the function becomes e*. To illustrate the method by which we now proceed to investigate the asymptotic expansion of b(«, @) when R(a)>0, we will first consider the particular case when 0=1. ees er ee) In this case od (2, Ne E(l+ 2h) ° T(m) P(r) , if m and n be both > 0, we have T (m+n) 1 Since | G=—aee™= dz= 0 a) yn pil $ (x, 1) =|) > Tm) y" (1 = yy" dy = | ay exp {xy (1 — y)} dy. 0 n=1 “0 Make now the substitution z=y(1—~y). We see that z vanishes when y=0 or y=1. It has a maximum value 1/4 when y = 1/2. If then we put 2=1/4— y=1/2+ 7, we have =7? and therefore n= + 01”. We take the positive sign in this relation when y>1/2, and the negative sign when y < 1/2. rl/4d im Hence od (a, 1)=| 5 (1/2 + 61) exp {a (1/4 — €)} €77 dé 8 £ ip 5 (1/2 — ") exp {a (1/4 —£)} 7 dg VC Woe) 1/4 = acess | ens Se 2 dg a4 = ter 2 | Evy 2 dy. 0 Suppose now that jargz|<7a. Draw a line J from 2/4 to oot, where |argt|<-7/2, this line not cutting the negative half of the real axis. Draw a line J parallel to this line from the origin. Then by Cauchy’s theorem co) (a, i!) = fers gi? [ (J) enaey? dy = dap? (1) etls-y mee dy 0 ai4 = de*t a? TP (1/2) -| (J) eY (1 + 4y/x)™ dy. 0 This formula shews us that, when |argz <7, p(w, 1) — ge** a P(1/2)), is finite for all values of |#|, however large. We can readily obtain a complete asymptotic expansion valid when |arga e4a® — 2 1.38... (2n —1) (— 2/z)”. - n=0 § 14. It may be remarked that this formula can be otherwise obtained. Jt ws a particular case of the asymptotic expansions of generalised hypergeometric functions. It LI (a\)= S rr aA 2", then, as I propose to shew in a subsequent paper, if | arga| "+ Jy, n=1 ! where | Jya#¥| tends to zero as || tends to infinity. Now ¢(a, 1) =F,,(#/4)—1, and we therefore have the expansion just obtained. We see that ¢ (a, 1) satisfies the differential equation 4ay"” —(a@— 2) y’-1—y=0. A second independent solution is, in the notation of generalised hypergeometric functions, ao 2 » F,{-1/2; 3/2; #/4}—1. § 15. We now proceed to apply the method used in the foregoing example to the general Function Malay & a" T(1+76) Sy 0). ee LO NEE Se 7) - Ee As before we have 1 $ (a, 0) = e{ yliexp cyl —2))| Cay renee coc secesetemacer eget (1). 0 We assume that y° is real and equal to exp {@ log 7}. Make now the substitution 2 P| (Unf) ora wears a sioiiouerscawisesseasis ie aoe a eee (1). Then z vanishes when y=0 or y=1: it has its maximum value 6°/(@ + 1)°" when y=6@(@+1). And z is real for all values of y within the range of integration. Put now 2=69/(6+1)4#—€; y=0/(@4+1)+7. * § 5 of the memoir to which reference is made in § 1. OF CERTAIN INTEGRAL FUNCTIONS. The relation (1) becomes 69/(6 +1)? — €=(0/(8 +1) +)? (1/(8 + 1) —n) g= allied Ean pees (@—1) 2 241" 134177" : n(@+1)| This relation may be written (eiks ayes " )"( 1 ae 2 (0 +1) ga-ga): gaea a "| =¢ ee pa 7 Hence, by Lagrange’s theorem n= > eno", n=1 6? fal \9 / ie ste —n/2 1 E (0+1)4 = ri + Jo \a@a1 n) where > =| .)— 3 n!|dn Ue n=0 This series will be convergent for sufficiently small values of |{|; say, when | 7) 0, we take the positive value of & in the series, and when <0, we take the negative value. We have dy dn On transforming the integral (1) by the substitution just employed we obtain P dy, (2 dys (a, 0)/x -f yi’ exp {xz} = dz +) oy! exp {az} = dz, ) where P= ope and y, is the value of y for which 7 <0, y, the value for which »>0 hs Ne 4 (Ge ) (hs = = Now st (galapn-2) [fpr -* and this when | £17) is sufficiently small can be expanded in ascending powers of |¢™ the positive value of /f being taken when 7>0 and the negative value when 7 <0. We therefore have, if | f7|0 o| 2¥2|_ 3 gene, yt| Ge |= ae ae / 6 \° (O41) as a= (555) V a0 The corresponding value of 4 a is obtained by changing the sion of V¢. 30—3 227 228 Mr BARNES, ON THE ASYMPTOTIC EXPANSION Hence od (2, )/x =i exp te (p = £)} = dn gna dt 0 n= + [exp le (p— £)} = (—)2 one dt 0 i as = Z| exp {a(p—£)} €7 {d,+d,C4+d,0° +...) dv. 0 If p>co, the series is divergent and must be replaced by the function from which it was obtained. 1 6 Bs * 5 : Wier io ee Now this function, which is the sum of the two values of P i+ @+1» ai is finite within the range of integration. The integral can therefore, by the theory previously established, be represented by the asymptotic expansion rs Vere TS | ens den cn-ss dé n=1-0 when R(z)> 0. If then R(«)>0 we have the asymptotic expansion f c iss don I’ (n — 1/2) , =?7 e | Pee) eS $ (a, 6) = 2a exp (@+D™= ee 7h ‘ The dominant term of this expansion is 6° ( QA 7 ) 12 exp \(0+1 yen Mt j \(e i 1)*2f The coefficients in the expansion can be written in a symbolical form by the use of Lagrange’s series. They can however be deduced from the coefficients in the asymptotic expansion of ee OE) Iya) pte | Glictan)ia © by means of the following theorem, § 16. If larga <3/2 and == j N (-y" if {n/(1 — a)} = J—a — L = = Tinal(l - Cee. ay O18 8) he > aaa aaa where |Jya%| can be made as smail as we please by taking |x| sufficiently large. + Jy, Consider the integral A | aie mea 27e J [(—as)sma(1—a)s’ taken round a contour which embraces the positive half of the real axis and encloses the points s=n and s=n/(l—a), Mi ON 12, we Oe We assume of course that 0—h. It is therefore equal to where | Jy2/"-*)| tends to zero as «| tends to infinity. If then arg a|< 37(1—a)/2, we have (1—a) f(a, a)—1— (29, 6)= (-)"T! n/(1 —a)} n= = If jan el — a)} a 1—a) +Jy. Change now # into a'~*, and we obtain the given result. § 17. Let us now apply this result to obtain the asymptotic expansion of (2, @), when R (a) >0. We have seen that, if jarg2 <(l—a)7/2, we have the asymptotic equality Flee a) = exp {d ae a) at (l—a) av a=a)} (ax)! 2(1—a) | S as (ax) a= ‘ 0 n= where a= T(Qn+1) |dr™| ar? ) =0; Tne | eee If then |argz|< 7/2, we have the asymptotic expansion ¢ (a, @) =exp {0%x/(@ ae 1)*7} | O1+8y nee as ov (1 + 0)**35 ee = 6 ? IiG@ee 1/2) ee 6 Or SO Bat ae} = | T=0 witere dn = (on + 1) [de | or Since d, = {27 (1+ 6)}", we see that the dominant term of this expansion is i Qo ‘ a+ ors)” exp {0%xr/(6 + 1)*7} and this is in agreement with the former result. 230 Mr BARNES, ON THE ASYMPTOTIC EXPANSION §18. It may be of interest to the reader if we actually obtain the asymptotic expansion of $(, @) when R(x)>0 for a particular value of 6. We can then determine the actual radius of convergence of the Lagrange series employed under the sign of integration. We will take the case when 0=2, and shew that, if R(«)>0, we have the asymptotic equality $ a” (27) ! ae 2 ese S ity (8n = 1/2) nar (en)! 9 pay Case Dee The function is represented by the integral 1 | y? exp (ry? —y)} dy We put y=2/3+7%, and make the substitution 4/27 — € = (2/3 +n) (1/3 —), or C=7 +17. The integral is then equal to 23+ Fed [Peslesnrtl a where 7, and 7, are the values of en correspond to € within the range of sooo {fel is-+mr integration. Now, if »=VE(1+7)%, we have 1(2 i 2 te qd? Ke 4 re 1 7! qpatt yy (y + 2/38) ee and therefore 5) dy ; = git 21 f qr a : (+1 ge 12 alee @ +23) | When 7>0, we take the positive value of /f and when 7<0, we take the negative value. On reduction we find that the above formula may be written P 3 e T (8n/2 — 2) 2 2 = See : nl2—1 7) CG ae oe T (rn) D (n/2) ae This series is evidently convergent if Guedes = <1 or |¢| < 4/27. Thus the series is convergent over precisely the range of values for € for which we integrate. We now have 2 (20)! Die ss [Pe 4 P (38n— 7/2) a —— 400/27 az OS n—3/2 nai_@n)! ip I, PARI = TV CEES cS OF CERTAIN INTEGRAL FUNCTIONS. 231 And, if R(z)>0, this integral is asymptotically equal to gain = ['(8n —7/2) qa LQn=h)\a 2 9 We thus have the given result. The first two terms of this expansion are = {4 Vrw Vara) ea ies 2 i It may be readily verified that these terms agree with those obtained from the general symbolical result of § 17. For, when 6 = 2, d, =1' (3) (4-7), = V6r, By ee als eee e SING) EEA 8 (ESTAS ae §19. The asymptotic expansion of p (a, 8) when R(x) < 0. We have seen that, if |arga#|<3/2 and @=a/(l1—4), we have the asymptotic equality 2 (=P T {ail —@) fh = pi-a _ . = Ss Se — Tope, o— Oe, 2) ae T Sna/(1 —a)| a” Also we deduce at once from the results of § 4, that, when |arg(—«)|<7/2, we have asymptotically 1 T {(n + 1)/a@} 1—a = see RS i Us f(a ? a) Gin=o iP (n + 1 ) (= a) (eH) O—2) [rig Ms the principal value of (—)"~*)* being taken. Therefore, when R(wv)<0, we have the asymptotic expansion en TS ee bye} % (-) T (n+ 78) CO Oo Pasta So haba | terms of order less than that of any negative algebraic power of |w| being neglected. We see then that, when R(#)<0, the function ¢(, @) needs two asymptotic series for its representation. There is, of course, no reason for surprise at this: it shews that o(#, @) is a function whose behaviour is slightly more complex than the functions previously discussed. § 20. We may obtain the previous result directly by considering the integral _ Ll patl+s) ~ QarvJ T (1 +84 Os) sin 3s round a contour consisting of that part of the great circle at infinity for which R(s)>—k, where k is positive, finite and as large as we please, and the line s=—k. If jarga|<7/2, we see that |Zx*| can be made as small as we please by taking |a| sufficiently large, when hk’ < k. 232 Mr BARNES, ON ASYMPTOTIC EXPANSION OF INTEGRAL FUNCTIONS. By considering the residues of the subject of integration at its poles within the contour we see that the integral is equal to & (— a)" T(1 + On) is (— a)" T (1 —né) i uy s eae on ean neo D{l+n+2é@n) nar UC(l—n—@n) 6 nao F (—n—(n 4+ 1)/@) C (n+ 1) sin (n+ 1) 27/8 ; A) arr 4cane 3 NORMs 2 Mean ecn Those terms are to be omitted from the summation for which respectively n> or (n+ 1)/O>k. We therefore have asymptotically, if | arg (—«)| < 7/2, x n 2 TD! 1 Ho Ded BT GH a OSs Pina Dea which is equivalent to the former result. § 21. The corresponding asymptotic expansions for the general function =. IN(QUS Gp) le Siege O00, g>0. Then if |argz|1/2, by the series of gamma functions <= T(pn—s)T (1+ an) mo: (1+ Bn)gh The latter method, by its analytical power, seems destined to play an important part in the further theory of expansions of more complex functions. X. A class of Integral Equations. By H. Bateman, B.A., Fellow of Trinity College, Cambridge. [Received March 22, 1906. Read April 30, 1906. Revised May 16, 1906.] I. Introduction. § 1. THE subject of Integral Equations bears the same relation to the Integral Calculus as the theory of Differential Equations does to the Differential Calculus, and the equations ; f(s)= | RAE OD EONS, oes: (1), F(s)=$(s)—”» ["e0s, AG (3) \e ice se paeenee aosenceoscionesacose (2), in which @ is the unknown function, present themselves just as naturally in calculations arising from physical problems as do the corresponding differential equations d =f@), F-r~w=f@), but strange to say until quite recently this branch of mathematics was practically neglected*. Now an integral equation retains many features of a problem which are lost when the description is given by means of differential equations, and the information which can be derived from it is as a rule fuller—the disadvantage is that the calculations are more ditticult to make. For instance if we replace an ordinary linear differential equation of the nth order by an integral equation of the form+ (2), we may obtain a solution in the form of a converging series of integrals which will give us full information about the solution at any point in the plane when the value of the function and its first (n—1) derivatives are known at a point which can be joined to the first point by means of a curve with continuous curvature, at every point of which the coefficients of the equation are finite and integrable. We thus obtain the information directly without having to resort to the method of continuation. * Abel, Liouville and Rouché called attention to the Soc. Ser. 2, Vol. 4, Part 2. new calculus and considerable progress was made later by + The necessary transformation is given in the paper Volterra and other Italian writers. A list of references will by the author just referred to. be found in a paper by the author, Proc. London Math. WG. XO ING, 3:6 31 234 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. In the present paper I shall be occupied in studying a certain type of integral relation between two functions h(s, t) and k(s,t). The general form of this relation is ad ‘i h(s, «) f(a, t) da = | TAS IMHO OYG. Srecbaoos9socqvonoese5960C8 (3), and it implies that the functions h(s, #) and k(y, t) possess a number of common properties which are of importance in the theory of integral equations. As the subject will probably be new to the reader I shall commence by recapitulating a few of the known results. It must be clearly understood that none of the matter which is given in the remainder of this section is original, but I have thought it necessary to introduce it here in order that the account of my own work which is given in Sections II and III may be intelligible. § 2. An integral equation is an immediate generalisation of a system of linear equations and important progress in the theory was at once made when it began to be studied from this point of view*. The linear equations (1) and (2) which are the simplest equations of this kind+ correspond to the system of linear equations n n p= & Krshe @=1, =m); Sr=$r-2% Krabs (P= 1), co) aes sone (4), and have been called integral equations of the first and second kind respectively}. The second equation is easier to deal with because the number of limear equations is equal to the number of unknowns and the known results for the system of linear equations apply almost word for word to the integral equation. Fredholm has in fact shown that the function ¢(t) can be uniquely determined except when X is a root of a certain whole function §() which corresponds to the determinant Nin, Kiss ee ; — XK, 1 —Dknn This solution can be expressed in the form b PO=FG)+2 | RGOFOM .....ce (5), an equation which is similar in form to the original one, a fact from which many deductions can be made. The function K(s,t) is called the solving function of the integral equation; it is a uniform function of X and can be expressed in the form A(A; s, t) * The new departure was made by Fredholm in 1900 by the variable s may be included in the ones given above (‘*Sur une nouvelle méthode pour la résolution du probléme by defining k(s, t) to be zero if t>s, a device due to de Dirichlet.” Oefversigt af kongl. vet. akad. Firh. Stock- Fredholm; the equations of this type have been studied by holm, 1900). Volterra. + The equations in which the upper limit b is replaced + Hilbert, Gitt. Nachr. 1904, Heft I. Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 235 where A(X; s,¢) and 6(A) are whole functions of X, their formal expressions being * A(A; s, t)=—k(s, t)+2A,(s, t)—d2A, (s, t)+... as SQQS T= Sar OAs= oa: ol [® (21/860, 46 %) &(3, a) i a6.0— 7 [-], b(s,,t),... (Sn, 85) If §(A)=0 the homogeneous equation ® 0=¥()-2] RAG MINER) ek Wy oh 3 Heat R ae eile (8), i rb ds,... ds;, 5, =a | Ana (s, $) ds. will possess a solution y(t) different from zero and conversely if a solution h(t) exists we must have 6(\)=0. Further the equation (2) will be impossible unless /(s) satisfies one or more relations of the form [Pox as=0. We have here a phenomenon similar to that of resonance, for if the function F(s) does not satisfy these conditions the solution of equation (2) will become very large as X approaches a root of 6(’)=0. It has been remarked by Fredholm that if k(¢,s) is written instead of k(s, t) the new solving function will be A (és) and the value of 8(d) will be unaltered, this shows that the equations 0=¥()-r] kG ty (t) at| ae Minera battens sear eeeddeae oats (9), C= paey at [a (8, t) x (8) as| can be satisfied for the same values of 2X. If X, is a p-fold root of 6(A4)=0, it can be shown that for values of \ very nearly equal to 2, K(s, t) eee TEN seis TROBE On 5s RR (10), where F'(s, t) is finite for X=2, and P(s, t) has the form P (8, t) = ry (8, An) Xr (6 An) + «Wp (8, An) Xp (E, An): The functions YW and y are linearly independent solutions of the equations (9) for this value of X and possess the important integral properties + ; 0 m#n | | whi (3, Xn) Xe (8; Xn) ds = . MUN, « UPI ba aso veeistinn conan (a): Ja l (=P, v= i) Also, in some cases the functions k(s, t) and K (s, t) can be represented by the series > Wp (s, Xn) Xp (t, >n)] D, i a8 ke ax yf state sToslolattarelaeibioutom erence sbiie ae (12), = pS; An) Xp\G An)! K(s,t)= = =e which are very useful for suggesting properties of the function K (s, ¢). * Tam following the notation given by Hilbert in Gétt. finite and integrable for the given range of values of s and t. Nachr. The functions k(s, t) and f(s) are supposed to be + See Fredholm, Acta Math. 1903. iG. = 31—2 236 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. We shall call the functions y,(s,,) the fundamental functions* and the equation §(A)=0 the determinantal equation of the integral equation (2), Many of the ideas that occur in the theory may be traced back to two papers by Poincaré +, where practically all the properties of the fundamental functions are predicted. The actual investigations have been carried out successfully by Hilbert who has established several important expansion theorems for the functions ¥, (s). § 3. An important advance made by Hilbert is the connection of the integral equation with the double integral fal» (S Dak(s) a Ode dt. wccescsere--- socuaah st eee (13), corresponding to the quadratic form Yksvsa,. The function k(s, t) is here supposed to be symmetrical in s and ¢ but this restriction is not of very great consequence; the salient point is that many of the properties of the integral equation are suggested by those of the quadratic form. The determinantal equation 6(A)=0 corresponds to the discriminant of the quadratic form and Sylvester’s theorem that all the roots of the discriminant are real remains true, finally the resolution of the quadratic form into the sum of a number of squares suggests the general expansion theorem from which so many other expansion theorems are deduced. When the double integral is positive for every function #(s) the fundamental functions vri(s), Wo(s)... may be obtained as the values of «(s) which make the double integral a maximum or minimum? subject to the condition i ie(e)hes =a, the theory being exactly analogous to that developed by Poincaré in the study of the problem of Dirichlet. The applications of the theory of the integral equation (2) are many and varied, at present the most important applications are to problems depending on finding solutions of linear differential equations to satisfy given boundary conditions. § 4. The theory of the integral equation b f=] ko bas is not so simple as that of equation (2) because there is no definite relation between the number of corresponding linear equations and the number of unknown quantities, In general in order that the solution may be possible it is necessary for the function f(s) to satisfy all the linear equations in s that are satisfied by the function /(s, ¢). * In Germany they are called the Eigenfunktionen, (1894). the function x (s, t) is called the Kern and the quantities \,, + The function y, (s) makes the double integral a maxi- the Eigenwerte of the integral equation. + “La méthode de Neumann et le probléme de Dirich- let.” Acta Math. Vol. xx., p. 120, 1897. ‘‘Sur les équa- introduced the next function y, (s) is obtained. The varia- tions de la physique mathématique.” Pal. Rend. t. v1. tional problem is ascribed by Hilbert to Gauss. b mum, when however the restriction i x (s) ¥,(s)ds=0 is a Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 237 For instance if d | x (s) k(s, t) dt =0, we should expect to have [rox (s)ds = 0. A complete determination of the necessary and sufficient conditions to be satisfied by the function f(s) will be a matter of very great difficulty. A method of obtaining the function ¢(¢) has been suggested by the author*, but its field of application has not yet been fully determined. It is important to distinguish two cases, first when a solution of the equation 0=[ ki ded exists, and secondly when no solution exists which is different from zero; in the first case the equation will be said to be special and in the second case general. Il. A certain functional relation. § 5. The study of a particular differential equation can often be facilitated by means of an appropriate transformation, accordingly it is natural to ask whether transformations can be profitably employed in the study of integral equations. The type of transformation which we shall consider here is of a peculiar nature as it can only be applied to a certain class of functions determined by the generating function of the transformation. The characteristic relation on which the theory depends is of the form b d axe a) f (a, t) de = | TEGLE GSB Gir sc tieeuecds eee (1), the function f(s,t) and the constants a, b, c, d are the elements of a transformation which associates a function h(s, x) with another function k (y, t). The properties of the function /&(y,¢) will depend partly on those of h(s,) and partly on those of f(z, t); we now inquire what are the properties that depend only on those of h(s, z). § 6. Let us suppose that the function /(s,¢) is such that no solutions (other than zero) of the equations = ie a(x)f (a, t)dz, 0 =| fe y) b(y) dy, exist, then we can prove that the quantities X for which the homogeneous integral equatious b 0=$()-r] h(a, #) (t) dt, d 0=x(a)-a] k (a, t)y (t) dt, Cc can be satisfied are the same. * Proc. London Math. Soc, Ser. 2, Vol. 4, Part 2. 238 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. Suppose for instance that A, is a quantity for which a function y,(x) can be found such that Fi Fs (@) =r | k (a, t) xn (t) dt, we form the function d ° 6, (s) =| Dy Cy Ye, 6 Cee gee ee sage tes (2). ad fd Then @n(s)=ru| | F(s,2) kw, t) xa(O) deat bra 3 =Aq [ | h(s, x) f (x, t) xn (6) dxdi, on account of relation (1), hence 8, (3) =n | f(s, 2) Oy (@) de Now @,(s) cannot be identically zero for the equation vd 0= | FS; ©) Xn (x) de is impessible by hypothesis, hence a function @,,(s) can be found to satisfy the homogeneous equation for X=A,. Moreover, if there are p linearly independent functions x, corresponding to this value of » the formula (2) will give us p linearly independent functions @,, for a linear relation between the functions 6, of the form a,@,=0 would imply the existence of a relation of the form ad 0= [f(s 2) Sanya(z) de, and this is impossible by hypothesis. Conversely, if A, is a quantity such that the equation b 0, (5) =e | h(s, 2) 0, (x) dz possesses a solution, then we know that for this value of a function ¢,(¢) also exists for which b ba(t)= An] W(S, 2)$a 0) ds o Now fet oe | fa) ise ae 3), b fe then Yale)=ra [| FO DAEs) bn @atds =X, a he k(s, x) f (t, 8) bn (t) dtds rad =r, | k(s, 2) vals) ds. Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 239 And since the determinant 6(A) is the same for k(s,¢) as for k(t,s) we conclude that a function y,(«#) exists for which a Xn (8) = nf k (s, 2) Xn (a) da. We can show as before that if there are p linearly independent functions @¢,, the formula (3) will give us p linearly independent functions y,. Now the number p is equal to the multiplicity of 2, considered as a root of the determinant 6,(\) of the “hk” equation, hence the root 2, occurs to the same multiplicity in each equation and we can enunciate the following theorem. Theorem I, If the function f(s, t) is such that the equations [ares t)b(t)=0 and ac t) a(s) ds =0, do not possess solutions, and if two functions h(s,t) and k(s,t) are connected by the relation b d | h(s, «)f(«, t)de= | FG, @) k (a, t) de, then the roots of the determinantal equations for the integral equations b d 0=9(0)-r{ h(s, «) (8) ds, 0=x(s)-r] k(s, @) x (#) de, are the same and occur to the same degree of multiplicity. § 7. The determinants will in general differ by an exponential factor, but in many eases they are the same; for instance if a rb F his, )=[Fe vay a)dy, ke, =] 9 (oe Nfly.t)ay, the equation (1) is satisfied and we can easily verify that the quantities 6(A) are the same. To do this, however, it is more convenient to use a formula for log 6(A) instead of that for 6(X), this formula is* — log 6(A) =a,r+ $a? + Lar? +..., rh pb where Qn = | Z| VA (Shs UK hp £59) 000 EX Gps GCA coal. Thus for the “hk” equation we have bra bod ee) ---| [Fo tg, Wf Cob) 9 (b &) on 1 (Gn9 CS: on Cag Cth one Chine and from the cyclical arrangement of the variables s,...sn, {...t,, we see that the quantity a, for the “k” equation will be the same. It must not be thought that the above formulae will give all the functions h(s, 2) and k(z, t) that are connected by equation (1) for this is not the case, the functions * Fredholm, Acta Math. 1903. 240 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. h(s, 2)=f(s, x), k(a, t)=f(a, t) providing an exception, for it is not in general possible to express the function f(s, «) by means of a definite integral of the form d fa=| Fo YoY ody. § 8. A particular case of some interest occurs when the limits (a, b) and (c, d) are the same and the function f(s, f) is equal to one when s is less than ¢ and eqnal to nothing when s is greater than ¢t. The relation (1) then takes the form “t “bh | h(s, x) de = | k (a, t) da. It is clear that we must have h(b, z)=0, and so we may write h(s, ®) =“ [9 (s, 2)-$( a) b and this gives $(s, t)-—G(b, th—(s, a) + 46(a, b)= | k (a, t) da. Differentiating with regard to s we obtain d HeN(Sstt) — = [$ (s, a) — $(s, t)]. It should be noticed however that this case is really included in the last, for a Rees h(s,a)=—[ ara b(n) dy, k(s, )=—[ b od (s, x). dz, and so we can verify as before that the determinants are the same. § 9. If the function f(s, t) does not satisfy the two conditions laid down we cannot assert that all the roots of the two determinantal equations are the same, because the functions @,(s) and ,(”) may be identically zero and then the above proof breaks down. We can however say that some of the roots are the same except in the extreme case when both sides of equation (1) are identically zero. It should be noticed that if a function w (¢) does exist for which [re t) w (t)dt = 0, then the relation (1) will still be satisfied if we replace k(a,t) by k(a, t)+o@(2) F(é. Also if we multiply equation (1) by w(¢) and integrate between ¢ and d we shall obtain |" “4s 2) k @ t) @ (t) di= 0. d The function a(x)= | k(w, t) w(t) dt is therefore either identically zero or a multiple of w(x); unless another function a(a) independent of w(x) exists for which “d | J (s, 2) a (a) dz = 0. In general the first alternative is true and so we see that the function /(a, ¢) in general satisfies the same linear relations in ¢ as the function / (a, ¢). Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 241 The fact that the equation (1) is satisfied by the two functions k (, t) and k (x, t)+@ (x) F(t) for the same function h suggests that there is some relation between the determinantal equations for these functions; accordingly we proceed to calculate them. d d eet | k(x, t)o(t)dt=po(z), [ o(2)F(a)de=e. d d Then a,=[ k(o,2)de, A,=[ {k(@,2)+o(a) F(@)}de=a,+0, d fd dd a= | [eek 2)ded, A, =) i {ke (aw, t) + (a) F(t)} {h(t, ) +o (t) F(a)} dedt = d+ 2eu+ Cc, and Ayn =n +(e +0)" — pb” Therefore —log A QQ) = E522 = ET UTN HH yn ee eee ye eo See we AO) 00) ee (A) = 8( E =a The determinant A (A) thus has the root F . F in place of the root ; possessed by 6(X). § 10. We shall now indicate another consequence of equation (1). Theorem II. If a(s) and b(¢) are two functions connected by the relation d a(s)= | f(s, t) b(t) dt, b the functions i ())\= [ h (s, x) a (2) dz, a B(ty=[ k (t, x) b (z) dz, are connected by a similar relation. " The proof is very simple, for if we substitute for a(#) in the expression for A, we obtain AG i } | " (s, x) f (a, t) b(t) deat =["[r x) k (ax, t) b(t) dudt Zz | ene ute If two functions h(s, 2) and k(a, t) have been determined and the solution of the integral equation for a particular function a(s) is known, then we may find the solutions for the system of functions obtained by repeating the operation é A (s)= | h(s, x) a(z) da, any number of times. WioTeXeXaee NOS exe) & 32 242 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. Again, suppose that we are required to solve the integral equation A(s)= [» (3, 2) a (2) de, the process may sometimes be simplified by first determining a function B such that AiG) | ac SV Ey then obtaining the function b from the equation a Bi)={ kG, x) b (a) da, and finally calculating a by means of the formula ACE il TC az) b (2) de. § 11. Theorem III. If h(s, «), k(a, t) is one pair of functions connected by the relation | a ESO [7 a) (a, t), de, hen i (s, =| ; f(s, 2)h (a, t)da, MOOS | f(s Eyl ees is another pair. e bd For | hy (s, t) f(t, 7) at = | FT (s, @) h(a, t) f(t, r) dtdx =|"|'r6 x) f (a, t) k(t, r) didzx = | eine § 12, Theorem IV. If H(s, x) and K(«, t) are the solving functions of the integral equations b F(s)=$(s)- r| h(s, x) } (a) da, d (8) = x()-r] kG, w) x («) de, then H (s, x) and K (a, t) are also connected by the relation [a GAYAG ndb= ["F6 x) K (a, t) dea. To prove this we recall the fact that H(s, ¢) is the solution of the integral equation corresponding to f(s)=h(s, t)*, so that we have the relation hie, Qe His, )— » [on (ei Maney: * Fredholm, Acta Math. 1903. Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 243 Multiplying by f(¢, r) and integrating between a and 6, we have i Go rydt = [a6 t) f(t, r) dt —r i i fo (s, «) H (2, t) f(t, r) dadt ......(A). Again, since A(z, t) is also the solving function of the equation FOLGE [‘ (aye alaa aahace Ela tata ean [‘% (a, t) K (t, r) dt. Multiplying by f(s, 2) and integrating ees and d, we have [ : FlecaLete aa | : ay Gua da [" [F(s, 2) k(a, t) K(t,r) dtde, but on er of equation (1) this may be written - 0 [‘ (a, t) f(t, rat [re aR Gh ey (i | “h (s, 2) f(«, t) K(t, r)dtde ...(B). Now the solution of the integral equation [Gof r)dt=8)-2 | 06,2)9 de, is unique, hence we conclude on comparing (A) and (B) that [a (a) fd, d= | a aioe oes This theorem completes our knowledge of the relations between the functions h and k, it shows us that when two functions h and k correspond to one another in the given transformation the whole system of solving functions for the different values of X also correspond to one another. We may sometimes use this fact to calculate the function K (#, r) when the corresponding function H (s, t) is known. § 13. As an example* we shall consider the equation | AG. a) dee (e peices 0 8 Ly pe La, (1 7 é) a, as E where el (Gi, 2) = (iene pe When s>&, we have [i (s, x) da =['a — £) Edz = q =~ E, 0 s so that h(s, €)= aah for s>&. 2 When s< &, we have & é L | h(s, #) de= | (1— &) x de+| (1 —2) Eda 0 8 & Ske) x Fa 0-0 E-9), (1—s)? 2 * I am indebted to the referee for the suggestion that examples of the theorems should be given. The actual construction of the examples has proyed instructive and led to new results. hence h(s, €)= +s8—€ for s&, we have ) sin {Wr (1 — z)} sin (EVX) ih Eke ae =| Vrsin VX ue é F [1 — cos {VX (1 — s)}] cos (EVA) which gives H (s, &)= Tan. and when s< &, we have & sin { sin {Vd (1 — &)} sin (w V2) sin {VA (1 — x)} sin (EV) ip ie aan I, Vr sin VA seis a VA sin VX ee _ sin {VrA(1— €)} (cos sVX — cos EVA) {1 — cos VrX(1 — &)} sin EV/r dsin VX rsin Vr _ sin VA (1 — &) cos sVA sin€VA_ 1 +—— = - =. sin VA AsinvA » T ; _ cos EVN cos VA cos VA (1 =e) bgrerore (a, §) Vysin VA Vr sin VX Hence we conclude that the solving function of the integral equation 1 F)=4()-Af HG Oat 0 We ye eu 22 SY in which a 2 JF +s—t if s Tekan An) x (E 1a)); but this may not be always true as there is some doubt about the convergence, moreover there may not be any obvious method of summing the series. If however we introduce a new function 1 (8, t) => — OS, An t, An), I(s, d) aa (8; Xn) X(t, Xn) where @(s, #,) are the fundamental functions for h(s,t) where the series h (s, t) =3 ~ 0(8, on) 6 (thn) is known to converge, and the quantities “, are chosen so that the series for /(s, t) converges rapidly: we may be able to determine the function / by means of the relation b d | OOS Un | Re meG ion. eee (i bis), To do this we must first calculate the integral on the left-hand side and try to find & by solving the integral equation of the first kind d TG | FG) GB 246 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. Let us suppose that the functions @(s,,) and also the functions y(t, %.) are linearly independent so that no solutions of d iF a(s) f(s, t)ds =0 and [ I (s, t) b(t) =0 exist ; then the function & as determined by the above equation will be such that the integral equation d 0=4()-a] k(s, t) ap (t) dt, is satisfied for the values (A, ...A,..-), this follows from Theorem I. for the equation b 0=6(s)-2| h(s, t) 0(t)dt, is satisfied for these values of 2X. Also the fundamental functions are known to satisfy the relations d 0 Ne #- An i v (¢, Xm) x (t, An) dt +. 1 Am = Nee ; 0 Xin # An “bh ip O(8, Am) (8, An)ds=1 yy: Hence if we multiply equation (1) by y(¢, X,) and integrate we obtain ada. d pb i | f(s, 2) (a, t) p(t, n) dedt= [| h(s, 2) fw, t) W(t, Xa) dedt b =|| h(s, x) O(a, X%») dx = @ (s, Xn) was li f(s, @) W (a, Xn) da. re ae > ot n 0 d Also the equation i f(s, x) @(x)dx=0 is only satisfied by w(x)=0, therefore we must have rd Xn i k (2, t) Ww (¢, 2) dt= w (a; Xn); and so the function (a, t) will possess the required properties. Il. A partial integral equation. § 15. The relation 6 d | H(s; a) Flas't) de= I FC Oo eee ee (1), may also be regarded as a partial integral equation to determine 7 when h and & are given, but we conclude at once from Theorem I. that it is useless to seek a solution except when the functions h(s,#) and k(a,t) are such that some of the roots of the corresponding determinantal equations are the same. Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 247 This condition is evidently satisfied if the limits are the same and we take h(s, t)=k(s, t); the equation i) b is ke (s, «) f(a, t) dx =[ EST ERO ee ee (2), is then closely connected with the integral equation of the second kind rb F()=O()-A] k HO Ode Theorem V. If f(s, t) and g(s,t) are two solutions of the equation, then a F(s, t) =| T(s, 2) 9 (a, t) da, is also a solution. 8 b fb var | GE Oe de= | k(s, x) f(a, y) g (y, t) da dy b pb =| i FT (s, x) k(a, y)g (y, t)dady a [- [Fe x) g(a, y)k(y, t)dady 6 =| Fo nk(y, tay, The function k(s,t) itself is evidently a solution of equation (2), accordingly the function b ere | ie Gai otic is also a solution and we may build up a succession of solutions in this way. § 16. Theorem VI. The solving function of the integral equation rb F()=4(9)-r]_k(, HHO at, is a solution of equation (2). This is a particular case of Theorem IV., it may be established at once however by comparing the relations k(s, t)=K (s, t)—r [xo, r) K (r, t) dr k(s, t)=K (s, t)— af Aalders § 17. Theorem VII. A function f(s, t) which satisfies equation (2) has in general the same fundamental functions as k(s, t). Let Yn (s =n | (5, 2) Fulv) de, . 4 . b then if [ K(s, 2) f(a, t)dz= : I (s, 2) k (a, t) da, 248 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. b pb b pb we have | | k(s, x) f (a, t) Wn (t) dadt -{ | FT (8, 2) k (a, t) Wn (t) dadt 1 rb =. | As, 2) vn (@) de Put Pn (s) = | FG. x) Wn (@) da, then bn ()=Aa f(s, 2) by (0) de Hence if X» is a simple root of the determinantal equation for k(s,t) we must have Kn dn (s) — Vn (s), b or Kn [ FT (s, 2) Vn (a) da — Wn (s), showing that w,(«#) is a fundamental function for f(s,z). The case of ~,=20 may be considered to be a degenerate case. If A, is a multiple root, this proof breaks down; but taking for simplicity the case in which there are two functions ¢, and y, associated with the root A,, we can either have Wr (s)= ya | £6 @) Wn (x) dx, dn (s) = mF 2) dn (2) da, or Wn (8) = Kn i F(S, %) bn (a) dx, Gn(8)=In i : F (8, &) Wn (x) da. In both cases the functions yw, and ¢, are fundamental for the function b GH)= | F (ep) f (et) de. § 18. As a particular case of this theorem we learn that if the equation (2) is satisfied by a function of the form FT (s, 1) =Wn (8) Xn (Os we must have simultaneously 6 wn (s) = ra | k (s, x) vn (x) dx, rb Xn (t) =An | Xn (x) k (a, t) dx, this is easily verified by direct substitution. We may build up a more general solution by adding together the products of this form for the different possible values of X; thus if the series > an vn (s) Xn (t) (Ss t), is uniformly convergent it will be a solution of equation (2), but it is not always necessary for it to be uniformly convergent. Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 249 § 19. It is natural to inquire whether any applications can be made of the results obtained with regard to equations (1) and (2). At present the most interesting occur when h (s,#) and k(s,¢) are the Green’s functions for linear differential equations; this ease will however be discussed in a paper shortly to be published by the author. We shall content ourselves here by considering another type of relation which leads to an equation of the form (2). § 20. Let us suppose that a function F'(s, x) exists such that b F(s,2+o)= | Ie SRG)ERE (EPR GES Foc ae. gaa cee hoe set vonsswet ge (3), for a certain range of values of w, then if W,(s) is the fundamental function for which b Wn (t) =Xn | k(s, t) Vn (s) ds, we should expect the function b gn (x) =| Wn (t) F(t, x) dt to possess a special property; as a matter of fact it appears to be a periodic function of the second kind. For bn (a ae @) = ks Vn (t) Fit, a+ @) dt = [. i vn (t) k (t, r) DR. a) dtdr b A |. Aen) (ey a — x gn (2). If we suppose that the function F(¢,.#) is given, the determination of the function k(s, t) will depend upon the solution of an integral equation of the first kind; this may or may not be possible but it is sufficient to know that there are cases in which it can be done. Let us assume that this can be done for more than one value of w, so that we have b F(s,e+o)=| k(s, t) F(t, 2) dt, F(s, eta=/ f( t) F(t, x) dt, and that these equations still hold when + or +o’ is written instead of « The two values of F'(s,z+@+o’) that are obtained are rb fb b rb i Heo DING ® Cup cad i | F(s, k(t 7) FO, @) dtdr, and these two integrals should be equal; hence if the function F(r,) is such that no b solution of | F (r, z)¢(r)dr=0 exists which is different from zero, we must have Db b i k(s, t) f(t, r) dt = | f(s, Dk (tr) dt, which is an equation of the type we have just studied. Vou. XX. No. X. 33 250 Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. Now we have seen that this equation implies in general that the functions k(s, t) and f(t, 7) have the same fundamental functions, hence we should expect that the function b bn(0)= | Val) Fito) at would satisfy the two equations Pn (x a0 w) a = gn (x), i 1 hn (@ + @') = —— gn (2). Hn There are now several possibilities. (1) ¢,(x) may be identically zero; this case is trivial. (2) The ratio of » and »' may not be purely real; in this case these equations are possible and ¢, is a doubly periodic function of the second kind. (3) @ and w’ may be multiples of the same quantity 0. (4) If @ and o’ are not of the form nQ and mQ these equations can only co- exist if $,(w) is of the form Ae~*, and in this case the quantities X, and p, are of the form e® and e respectively. § 21. For purposes of illustration it is convenient to consider a particular example, accordingly we shall endeavour to determine a function /(s, ¢t) such that 1 =|" k (s, t) dt bien 1 2G o) Gel a wee) lee on the supposition that #>1 and that is positive. Writing o=5 the equation becomes sling eer a k(s, t)dt pe PC —foree) 26 Soy lt | aoe Now the solution of the equation , a +1 _ p(t)dt 18 given y b= 2 [t+ VIP f+ VI #)— (t- iV -B) Ft - iT 8} Hence we obtain (omitting the algebra) 1 1—2s(a@+o)+(@+o) oe +1 dt o (@— 2s)V1—# a} 1—2ta + a2" 4(t—s+ 0) + 4t(t—s + @) (@? — 2s) + (@? — 2sw)?’ 2 9) 3 so that k(s, t)=— @ (w — 2s) V1 —¢ 7 4(t—s+ ow)? + 4t(t—s+ @) (w? — 2sw) + (w* — 28a)?” * This result will be proved in a paper shortly to be published by the author. If ~ is >1 it is sufficient to change the sign of ¢ in this formula. Mr BATEMAN, ON A CLASS OF INTEGRAL EQUATIONS. 251 If now we give a different value we shall obtain another function f(s, t) which will be connected with /(s, t) by the relation (2). Writing the above equation in the form H F(s, r+o)=| k(s, t) F(t, x) dt, —1 41 ti F (s, 7+ 2a) =) | k(s, t)k (t, r) F(x, x) dtdr —1/-1 +1 =} k, (s, r) F (r, x) dr, —1 F(s, 2+no) = Ih Ken (Ss, 7) (ir; a) dr: J—1 we have and But for small values of X the solving function K(s, t) may be expanded in the form * K(s, t)=k(s, t) +k, (s, t) +... Aken (5, 1) +--.; +1 +1 where Kenn =| Sco || PACA AVE (Gig ED) cog Lal Ga, CEA Ben Se Hence if G(s, c)=F(s,c+o)+AF(s, 7+ 20)+..., oa we shall have G(s, x) = K(s, t) F(t, x) dt. 1 1 1 1 3 : == N —_ = _ —— —Z(y—8) | MT 2 ¢l f a 1—2sy + y= 1—s*+(y—sy al one ro provided s*<1 and y>1. Hence > .—~ Me 2 INS “etletnenn sinzV1—s?.dz 1 1—2s(e@+nw)+(e@+no)? VJ—s7 2 : 1 Ge e72(ttw—s) as 1 gaz Sinz V1 — 8°. dz, Se/9 US [FAs le - |" K (s,s) ds. Putting s=cos 6, we have i K (s, s)ds = —= zai | alse = sin (z sin @) sin (z sin 6 — 6) dzd@ { To Jo 1—re es : 2 a -={" ip Paver [— cos 6 + cos (22 sin @ — @)|dzd@ eo =i i= er (22) dz. ee hee! K Now fe (2) =5| 1-7, 6 (A) 2 | (n+1)o | Sa dS) ~o Vo? (n+1)+4 The roots of the function 6(X) are of the form e*” for we have S he, eee 1 A” V2 2@? + 4° and this is satisfied for all values of n if X; is of the form e%*. Now this is exactly what was predicted at the end of § 20, accordingly we can say @ priori that the fundamental functions are such that ae / 1 vr (t) dt ae , l—2te+2° and as we have just seen, the corresponding value of y is DAT : —{. —— , +0) —— =e eit [tsin p; V1 —2&—V1—# cos p; V1 — #). The quantities ¢*% are the roots of the function §(A) and are known to satisfy the relations S als = S e7rop,; — il | = ue) = | : 1A" 2 V n2w? + 4 The problem of finding the roots of 8(A) is therefore reduced to that of expressing the function : E - 5 by means of a series of the form - Vae+4 for positive values of a. XI. On Functions defined by By E. W. Barnss, M.A., [Received ¢ § 1. IN the present paper it some simple cases of integral func metric series. The general type of such seri a,(a, +1). Chenu: pe tee : * 1.2 p,(p,+ Wore. Po a+ Such a series is an integ The series we shall denc or, more briefly, when the The series satisfies th ie + a> wherein $ der *~- The and (q-1, From one , functions is that of . of z=0, and of com WoL xox NGO. X DEFINED BY =1 has been undertaken by Jacobs- Orr +, who in two papers suggested ssults by laborious ductive analysis. elementary cases of the theory in a nvolving gamma functions, and of the general investigation on the Asymptotic *s Series. In a paper with this title § igh no detailed investigation was given, ‘s by the author‘). Complete references in the paper first cited. a complete discussion of the function uded in the general category. 43 p,—a}. nptotic expansion of ,F, {a; p; a} is previous results gives the asymptotic x of the theory then follows simply. iis function is substantially Bessel’s of ,F,{a; p; «} by means of the {op —1/2; 2p —1; — 4}. n obtained, and will shortly be ietric type ‘ -» ineluded). If ol. XXXVII. pp. hematical Society ambridge Philosophical -232. Quarterly Jowrnal of 108—116 and pp. 116—140. 2 SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 255 The series satisfies the equation * ; d \s +a)— da +p — Ly y=0 _ ey i dy A °: 2 ale Cre ckeie Gan An independent solution of this equation is we 1B, (a—p+1; 2—p; 2). § 5. We will shew in the first place that, for all values of # of finite modulus, Fi {a; p; a} =e Fi {p—a; p; —a}. Consider the integral z = Da syeds round a contour C embracing the positive half of the real axis and enclosing the origin. By Cauchy's theory of residues it is equal to &, ( — yn germ = nl Ce Ten m=0 me. Ei(a@)n= 2 V(a+n)e*%a Tow ee ey fa: : gt >> r Ly ' ae P(p)° a de n=o0 V(p +n) (n+ 1) Assume now that R(p—a)>e>0. Then the last series is equal to 1 = Tia+n)U@=s) 2re) ¢ nav V(p +n) C(n4+1)' For the series under the integral sign when divided by I'(—s) is convergent if sds. R(p-—a+s)>0, that is to say at all points on the contour of integration, if the contour passes very near the origin. And the integral is equal to the sum of the residues of the subject of inte- gration inside the contour. But by Gauss’ formula = T(at+n)U(n—s) T(p) nao V(p +n) UC (n+ 1) Pa) P(—s) =J1(Ch Soe os 1), ['(p) Ul (p—a+ts) ‘dea rh 7 —at+s)>0. TYSON SES provide (p—a+s)> We therefore have RG pie. poe 1 PRES ee me Dey 5 Ep)” {a5 p; z= 2m le T(p—2) F(p+s) as al) (—)?2" T(p—-a—n) ~ Pip—a) nao Uw +1) (p—xn) by Cauchy’s theory of residues D(a) p, =F hile—3 p; — 7}. * The equation has been considered by Pochhammer, Mathematische Annalen, Bd. xuv1. pp. 584—590. 34—2 256 Mr BARNES, ON FUNCTIONS DEFINED BY Thus, if R(p—a)>e>0, we have the required identity iF, fa; p; s}=e.Fi {p—a; p; —2t}. Since each side is continuous as a function of w for all values of @ and p such that p is not zero or a negative integer we see that the theorem is true in general. The important result of the present paragraph appears to be due to Orr*. A particular case had been previously obtained by Glaisher*. As the method is one which will be constantly used, we will indicate a modified form ot proof. § 6. We know that, when x is very large, R@amhG@—s) fay Cy cr , JR(w Dip +n) (n+1) nest ite Spicy aoe eel the c’s being assignable polynomials in p, a and s, and Jp(n) being such that | Jg(n)!| can be made as small as we please by taking x sufficiently large. 1 2 (TiatnTin—s) 2 Crm) ano) ~ Saye ce R(a—p)—R. For such values of s that R(s) > R(a—p) it is equal to T(p—a+s)V(@) 2 ¢€(p—at+r+1+s) T(p—a)V(pt+s) +20 P(—s) , where £(s) denotes the simple Riemann € function. This equality between analytic functions therefore holds provided R(s) > R(a—p)—R. And by taking R as large as we please but finite we see that it will hold for all values of s such that R(s)>a finite negative quantity. Consider now the integral ga (M(p—ats)T(a)P(-s)_ =, yl, te T'(p —2) (pts) at BAP 2 ee ea i taken round ‘a closed contour Cy which encloses the points s=0, 1, 2, ... V, and excludes the points a—p, a—p—1, ... which are poles of the Riemann €¢ functions. It is by Cauchy’s theory of residues equal to Pia) P(p) the suffix N denoting that only the first (V¥+1) terms of the series for ,F,{p—a; p; —a} are taken. Pi {p—a: p; — «hyn, * Orr, Cambridge Philosophical Transactions, Vol. xvu. + Glaisher, Philosophical Transactions Royal Society, p. 175. (1881) Vol. 172, p. 774. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 257 But it is also equal to 27 Jc pe ahs) = (har R@ss)e) 2. ane 2 pal T'(p) parent (P(ptryC(n+1) Zo ne attr! ages La a = x (C(at+n)T(r—s) aa Gy nzole (oe ere Ge 1) y (a ee n) v= n (—)yr gitm n=0 T(p+n)C(n+1) x m=0 : m! We therefore have P (a) a wieiss (a+ n)(— ya m+n D(p)’ ee VES SS NES yl (aoa Cpe ape all values of » and m being taken which are positive or zero and such that m+nN have evidently a sum which tends to zero as NV tends to infinity. The double summation is absolutely convergent, and may be summed in any way we please, I (a) Phd, Fie T(a+n)a” ine RGA Mid ie ee Gee DG or iP, {a; p; 4}=e Fi {p—a; p; —a} § 7. We will next shew that, when R(x) <0, ,F,{a; p; 2} admits the asymptotic 1) (— «)-* .F, |e 1p +a: aly: expansion [ (p) P (p —4) The error which results from stopping at the kth term of this series is at most of order a~***? when || is large, and (—«)-*= exp {—alog(—~)}, the logarithm having its principal value whose imaginary part lies between + 77/2 Consider the integral 1 (Eee P(ats)(—2) 5, Q7r. I (p +s) taken round a contour enclosing the poles of IT (—s) and no other poles of the subject of integration. In the integral (—z)* has its principal value. The integral is evidently equal to 2 P@rn) # EN) 7, ca). ne =u CSE Bey) Dp) Now, if R(#)<0 so that jarg(—2z) <=, the integral vanishes when taken round 2 ? that part of the circle at infinity for which R(s) >— hk. 258 Mr BARNES, ON FUNCTIONS DEFINED BY Hence asymptotically the integral is equal to s (Mes eaaee r-o Vint 1)T(p—a—n)’ : sl : : terms of order less than that of any algebraical power of , when |) is large, being & = neglected. The series may be written = Pa+tn)T(a—p+1+n) sinx(p —a) — ~\-2 > is Sees Tatl(—2 7 Ae (es eee Ae | =(—2) Toone Aes pta; Thus, if R(z)<0, 5 AA ke STE aa lie soe if is ral RES Pah ae) Oo ab 1 AOR eas §%. Consider now the nature of the asymptotic value, when |@) is large, and R(«) <0, of .F, ja; p; } when a and p tend to positive infinity, in such a manner that Lt (a/p)=1. We have, when R (x) < 0, i Py {2} =- Q7 | T(a+s) T(p) T'(a) DT(p+s) [r (— s)(— a and, however large |a and |p| may be, we may take this integral along a contour in the finite part of the plane parallel to the imaginary axis so that —a is to the left of the contour. If then & be any finite quantity, and a and p tend to positive infinity in such a manner that Lt(@/p)=1, Ee (x) ae te F, iz} = yes > where, however large & may be, we can make |J(#)|, when || is. sufficiently large, as small as we please. § 9. We may next shew that, if R(x) >0. e I'(p) | ‘| a 75 orl Ny Se eeae Eats Ps) (a) are \P a a| The error obtamed in stopping at the kth term of the series is, when || is large, at e= most of order paces To prove this theorem we combine the two previous results. Then we see that if R(«) <0, &\F, {p—a; p: —a} is asymptotically equal to T'(p) Tip—a) Change « into —«, p—a into a and therefore a into p—a, and we have the result stated. Dav ch, {o, l—p+ a: -. i SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 259 Combining this result with that just obtaimed we see that, if | arga| <7, s ZIG a 1) Ff, {a3 p; B= pq eet pp % os @is af A a ae nara a) aly a l—-p+a; - +. This is the complete asymptotic expansion of ,F,{a; p; #} in the vicinity of |a|=2, Tt shews us that the large zeros of ,F, |a; p; “} ultimately lie along the imaginary axis. § 10. The nature of the asymptotic value of ,/, |; p; 2} when || is large and f(z) >0, when a and p both tend to positive infinity in such a way that Lt(1—a/p) tends to unity, can now be immediately deduced For, when R(x) > 0, if, {a; p; 2} =e Fi |p—a; p; —at. Hence if & be any finite quantity, and p and « tend to positive infinity in such a way that Lt(a/p) tends to zero ets) a v : « ml iF, {a3 p; x} where, however large & may be, we can make |J(#)|, when || is sufficiently large, as small as we please. § 11. We have seen that the fundamental differential equation d* dy aL daz — (rf — p) aa = ay = \\)) admits the two solutions Ff {a; p; #} and Fi ja—p+1; 2—p; a}a'. The idea therefore suggests itself that it is possible to take such a linear combination of these two imtegral functions as will admit all round =x the single asymptotic expansion 1! (—«)* .F, ie l—pta; — and that equally we can take another combination which will admit the single asymptotic expansion 1) e 27-? FB iP —a, l—a; A f We proceed to prove that such expectations are correct. We will first shew that*, if |arga) < am. T (a) Q—p),F, fa; p; 2} +T(a+1—p)P(p—1) 2" PF, (a—p +1; 2-p; 2) =[(a)TA+4—p)a*.F, ie 1+a—p; a * Compare Orr, loc. cit. p. 178. 260 Mr BARNES, ON FUNCTIONS DEFINED BY Consider the integral 1 aa 27 | T(—-s)TU—p—s)T(at+s)ads taken round that part of the circle at infinity for which R(s)>—k, and the line s=—k. The integral along the circle will vanish if ee OTe jarg Z|} < > . 2 : ere. ae 1 The integral along the straight line is of order jel Hence we see that = (-) te’ r i r > a (l—p—7n) (a+) n=0 : =) (— y2—) gi-pta F + > ————— P(p—-2v-1)T(a+1l—p+n) n=0 7. : d (= al . + 3 ¢*7+—T(a+n)T(1+a+n-—p) n=0 nr: =r, where R(a)+1+1>hk>R(a)+/ and J; is, when 7) is large, of order = Thus asymptotically, when argw < aa : T(a)T(1—p).F, fa; p; ec} +P (a+1—p)T(p—1)2'.F, {a—p+1; 2—p; a} a2 Fa, 1+a—p; —tr(@Ta+a—p) (A) ( < § 12. The previous result has been established when |arga|<3r. Therefore when R(z)<0 it is equivalent to two different results. And by eliminating the function WF, \e-P+ ils Fi 93 -i we can obtain the asymptotic expansion of ,F, {a; p; 2} when R(«)<0. This is readily done as follows. Let l(a) (1 —p).F, fa; p; =P, T(a+1l—p)T(p—-1),F, {a -—p+1; 2-p; z}=Q [ (a) (1 +a—p).F, \n, lia-p; -it=R and suppose that arga lies between $7 and r. Then the previous result is equivalent to the two relations P+2-°Q=a~— R, P + gi-p e727 (1p) Q =a2-8 em R. the principal values of z~? and «~* being taken. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 261 Hence P(e"? —1)=a-* R (e7? — oe), or Psin rp =e" 2-* R sin 7 (p — @) =(—«)~* Rsin r(p — a), where (— z)~= has its principal value, the logarithm by which it is defined being real when @ is real and negative and having a cross-cut along the positive half of the real axis. Similarly, if arg@ hes between -2 and —7, we shall have P sin rp =e** «* R sin r(p—2@). Thus in either case Psin tp =(—«)-* Rsin 7 (p— 2). Hence, if R(x) <0, T (a) T (2) 1) Sy: : = ——-* (a ee =F SE Ras) Ps “lr () RG@=a. om) Pa 14a aaa or TAG Cp Rah — a he eat a, l+a-—p;—- “| ~ Tip We have thus obtained the previous result. § 13. From the result (A) of the last paragraph but one, we can immediately shew that, if | arg(—2)|< 2 ; ra paar — =F, {a; p; a} +reapco ifa—p +1; 2—p; a} = e&(—z)2- Ae \p — 4, 1 wo . For putting p—a for a in the result (A) we have, if |argz|< = Ff, |p—a; p; e+ F, {l—a; 2—p; a} at T'(1 —a) T'(p—a)’ 1} eee Ne lo—a, l—a: ial , Now Fi {p—a; p; z} =e7.F, fa; p; — and if, {l—a; 2—p; cz} =e, F, {l1—p+a; 2—p; —a}. Thus, if jarg a |<, we have Ep) T(p- 1) = f 0 eae i RSs) yf (a; p: Zia) aa °F {l—p+a: 2—p;: —a} 1 =_ po 6 — — '—--—>, =e: | a, 1—a; A Changing x into —# we have the given result. When R&(x)>0, this result is equivalent to two different relations. By eliminating the function ,F,{a—p+1; 2— p; 2} we can obtain the asymptotic expansion previously given for ,F, {a; p; 2} when R(a) >0. Vio. XeXS Now xe 35 262 Mr BARNES, ON FUNCTIONS DEFINED BY § 14. We proceed to give a direct proof of the asymptotic equality of § 13. *a-p-2 -a-p-l -a-p Let C, be a contour as in the figure which encloses the points Qae Se 1—p, 2—p, 3-—A, ..., and excludes the points a—p, a—p—1, a—p—2, ..., and which is that part of a contour embracing the real axis which is included within a circle of radius » together with an are of this circle. 1 3 / T(-s+#)l(1—p+t-s) "Pmt soto, TEPDIG—a—s42). te Consider It is equal to 1 AK st) — pes) eas | T(—s+t#)T(l1—p+é-—s) . ————— SN = = Sli P(t+1)PQ—-a—s+t) ds 2mevzenJo, TE+1TA—a—std) pe 1 N31 (T(-s+dP(U-—pt+t-s) 2 «(s) )., Pee tarsi mresra - ads ae fste—atrs+i | 2 Qa | iS | T(—s)l(1+p-s) attt de, Qru=w Jo,_,T (+1) TA-4-s) R the double accent denoting that the summation = does not exist when t=0. The r=0 coefficients c¢,(s) are analogous to the polynomials introduced in § 6, and are defined by the equality (1) below. The integral is thus equal to a = (iieet)l dps) Ge) | de Orel oc, 20 (PEF EO =a ee8) | pce ay” AL Se eas T(—s)lQ+p-s). : Qa Corrine T(l-—a-s) pate mk : ily 2 (Pesto Garret) Ss eee Ss where SSS Ih JylEG4+D PO -assth) ~ pp ieernl Oe. Now, if ¢/A is large, Ne Gee Te) ere) Tpit) _ a TG+DPGsa—stt) pe EH ) where Jp(¢)| can be made as small as we please, by taking ¢ sufficiently large, for all finite values of R. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 263 f S Jn (t) re ds. Hence ty = — > = 3 N Qa G, t=N fste—a+R+1 And therefore, since we can always take N so large that | Jp(s, t)|<» if t>N and t/X is large, m being a positive quantity as small as we please, we have hee Se | in| < | | a ahE 27 | Cy t=N | {stp—a+R+1 | ds |. Making WV tend to infinity and A also tend to infinity in such a way that r/N is small we see that | Jy| tends to zero, if R be a sufficiently large finite positive integer. Again, when WV is very much larger than 2d, the terms of the series wo at Eigse p38) a. Cy-t Sees Nie S7)B ENS t-v (t+ 1) T(—a-s) all vanish. Hence if C be the limit of the contour C, when 2 tends to infinity, we have ie / T(—s+t)l0d—p+t-—s) SEP ol URE AIIM TSC rer eae — 1 E = [ S(s)— = ,(#) (e+ p—atr+1)| ads, Cc r=0 4 ~ Qare | where S(s) is the function defined when R(s)> R(a—p) by the series = P(—s+t)Td-—p+t—s) oI) p= Se ke Thus, since €(s +p—%+7+1) has no poles within the contour C, we have U ee gees tae SRG : Gk aes) wds=—s—| S(s)atds. § 15. We have incidentally seen that R 1 as a . < [se ice) Ela p a-r+))| ese can be represented by an absolutely convergent series valid when R(s) > R(a—p)— R. Thus the sole finite singularities of S(s) are poles at the points a—p,a—p—l,a—p-—2, ...... And, if R(s)>R(a—p)— AR, and the coefticients a, be suitably determined, S(s) i a, aS hla Sali, (Sasa 5) sansa a ee F(s), where F’(s) is an integral function. By Gauss’ theorem we know that S(s) ['(s+p—a) T(—s)T(—s+1—p) T-a)T(p—a2)’ 35— bo 264 Mr BARNES, ON FUNCTIONS DEFINED BY We thus verify the previous result and shew that Bb! g r!TQ—a)T(p—a)’ a We now have Re So ete a pe a fp SON Eee Qrito T(1—a-s) (1 —a)T(p—a) The first integral is equal to the sum of the residues of the subject of integration inside the contour C. It is therefore equal to wo =p = (SP yng 2 (— re ip = em) = ===) n! ea n!T(p—a—n) a iG) fi) T(p—-1) _ qi ane rg? re ee The second integral is zero along a contour at infinity for which the real part of s is greater than a finite negative quantity, provided | arg | < 37/2. Hence it is equal to of § aE ye eee) n+, =o 7! Tl —a)T(p—a) where |J;| tends to zero as |#| tends to infinity. It is thus asymptotically equal to at-?.F, \p—a, l—-a; -7. ( 2 Finally te if | arg (— a | < 37/2, Tp) aes Te DN (pe ) F, {a= ps T= oo °F {a—p+1; 2-p; z} 1 =o (a) Fy tp — a l-—a; ne § 16. To verify the complete asymptotic expansion for ,F,(@; p; 2), which was given in § 9, we will obtain the result directly by the use of Pochhammer’s contours and their deformation. By his result* we know that for all values of p and q P(p)l@ 1 T(p+q) (—e™?)(1—e™9)’ the integral beimg taken round the contour P of the figure. i—~ = : Se ih * Pochhammer, Mathematische Annalen, Ba. xxxv. pp. 470—526. | vPA(1—v)3dv= JP SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 265 The argument of v?(1—v)!~ on the initial line is zero. Then we make a positive circuit round 1, so that the argument of the return line is e&%. Then follows a positive circuit of the point 0, so that the argument on the following direct line is e'?+9, Then we have a negative circuit of the point 1, so that the argument on the second return le is e”, Finally a negative circuit of the point 0 brings us back to the starting- point with zero argument. (ere er ee eed OE ie aor ees L(y <=) DG) Case) ) 1 (aig (—a2)" = —— = > ae a-n—l = j—a—] DT (p —a) (@"* — 1) (e -@) =e Evpgrennh ieee iT = = Tip <= a) (ere — 1) (e (p—a) __ Dl é where p=a, y=p—a. 20 yP1 (1 — »)@" do, Suppose that R(x)>0. Then we may modify the Pochhammer contour till it consists of the four contours of the figure. On the first contour the argument of v?"(1— vv)? is initially e~™4~ and finally em (q-)_, On the second contour the argument is initially e*"—~). After the semi-circle round the point 1 it is e, After the loop round the origin it is e+, and finally bp IS ere Oz Paes) On the third contour the argument is initially e«~?~, and finally, after the negative loop round the point 1, it is e™ 2), On the fourth contour the argument is initially e* "2. After the positive semi- circle round the point 1, it is e™’). After the loop round the origin it is e™, and finally it is e~™* 9, 266 Mr BARNES, ON FUNCTIONS DEFINED BY If now we add together the integrals round the four contours we obtain [, e-*” yP—-1 (1 — 9 dy =(1—e&™?) [ezuen (1 + vj? (— v)2" dv +e" (1 — 82) | e- (—v)P1(1— v2 da, the integrals being taken round the usual contour G for the gamma function deformed in the second case so as to pass above the point v=1, and (—v)?" and (—v)? having their principal values which are real when v is real and negative. Suppose now that arg lies between _7 and 0. Then by the substitution av=w a an\P—1 / q-1 , / oo ( i ®) fe “) ay J x \ x x round the new contour of the figure. the second integral becomes (= f wi : ; Suppose now that we expand (1- =a by the summable divergent series ZL / Sc etal any (eae nao U(1—q) UT (n +1) \a/ This will be valid in the plane dissected by the line passing from # in a direction : ae away from the origin to infinity. Therefore when this series replaces (1 - ) under the integral sign, the value of the imtegral along the w contour will be equal to that of the integral along the contour G which embraces the real axis if arg lies between -5 and 0, which is the case represented in the figure. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 267 Similarly in the first integral we may employ the summable divergent series for (1+ »)?>, though im this case since R(x) >0 no such difficulties arise. Therefore | e-2" yP— (1 — v)9"' dv Jp : z T(n+1—p) Q7 — — g2m —«S NE a : ee yer A aay ee © Q—n—9) Tl (n+1— q)(-)” 2a mp — emg S = 2+ Ge >, Peg) R= ya =* ane)’ the principal values of w"*? and 2‘? being taken, and the series being asymptotic in Poincaré’s sense. Thus if R(x)>0 and if also arg lies between = and 0, ey. , EP@E@=—a)n Se ik oe , (n+1—p)C(nr+q) GE Lay eas EGOS ani aa Dd —p)l(n+)) a = Pvt+p)T@+1—- q) a-v (1 —g) (rn 4+1)a™’ ; . : Crk p—a) T (p) ma / as : -*) op G3 p32) > aye ell p85 2 T'(p) ee) — _,U(p) ale ( = = 1 =e T(a) | 2) * ay (1 a, p—a; -;) I (p) a: t+aTG cai (* 1-e+a: 5), (1) where (— z)*~* has its principal value whose argument lies between }7(a—p) and +(a—p). The reader will readily supply the deformation which it is convenient to take for the original Pochhammer contour in the case when argwz lies between 0 and da. By it we establish that the formula (1) is valid whenever R(x) > 0. Utilise now the identity ,F, {a; p; 2} =e*,F,{p—4a; p; —2} or if it is preferred modify again the contours of the original Pochhammer circuit and we establish that asymptotically in iP, (a; p; a= eae P,(p=4, Le; ;) T'(p) fee arrit ; *) (— 2) fi (1 p+4, a; al whether R(x) be positive or negative. In this equality (—«)-* has its argument between —ar and a7, and zt? has its argument between —(z—p)7 and (a—p) 7. This is the result previously obtained. + See § 5 of the first memoir by the author quoted in § 1 of the present paper. 268 Mr BARNES, ON FUNCTIONS DEFINED BY § 17. We have now completed the discussion of the asymptotic behaviour of the function ,F, fa; p; a}. But before proceeding to other investigations we may note the connection between our analysis and the theory of linear differential equations. We saw that iF, {a; p; a} and «Ff, {a—p+1; 2—p; 2} were the two independent solutions of the linear differential equation d*y _ dy © Te (e—p)4 ay = 0. da Hence from our results we also see that 1 a? F.ia, 1+a—p; ari ( 1 and Ca tome. Alias P —a, l-—a yoo t ) av are two independent asymptotic solutions of the type first introduced to analysis by Poincaré. These asymptotic solutions for this particular equation have been obtained by Jacobsthal*. If the associated series were convergent one integral would be regular and the other normal. ow an I'(p) The function F, (x)= a T(ptn): § 18. If we put a=1 in the asymptotic expansion which has been obtained for iF, fa: p; x} we obtain F,, (x) = ex'-* T' (p) oF, \p-1 0; I +(p—1)(—2)7 Ff) l2 —p,1;- i: In this equality «° is such that |arga|<7, that is to say, a has its principal value with respect to a cross-cut along the negative half of the real axis. : iL Evidently Ff, fs —1, 0; | =1, { x 1 2 (p—2)(p—3)...(p—l—n and F,j2—p, 1; ~jt=14 3 ©- 9 Beeke Me v n=1 x Hence, if | arg «|<, we have the asymptotic equality N (p—1)...(ep—n F, (@) —e #'-° T (p) =— SS (p = Se (A) n=1 v where |Jya%| tends to zero as |x| tends to infinity. We can immediately verify this result when p=1 or 2. lips For F, (~) =e? when p =1, and F, (2) = (e*—1) when p=2. * Jacobsthal, Mathematische Annalen, Vol. Lvi. pp. + Forsyth, Linear Differential Equations [Theory of 129—154. Differential Equations, Part ur. Vol. 1v.], Chapter vi. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 269 § 19. The previous asymptotic expansion (A) can be readily obtained de novo as follows, when R (p) > 0. =~ a T(n+1)T(p) Tr ~ > = rans ae aa Te We have EMC Se Sal ranma e a seis = i a" (1—y)" n=0/ 0 n! ye dy 1 = 1+ xe” / ey ye dy. “0 The process of summation under the sign of integration is justified by the fact that F,(x) is uniformly convergent for all finite values of | «|. x Hence F, (a) =1+ &a'- | en? dn. ~ 0 From the origin and the point # respectively draw two parallel lines A and B to infinity making an acute angle with the positive direction of the real axis. When R(«x)<0, the line from z must not cut the negative half of the real axis. Then [ + |= Jn ; “0 Jz “0 Hence, if |arga2|< 7, F,(a)=1+ea' | es en? dn — jw Canine dry\ “0 7 =1+éea'?I(p)—e@ a'r I: B e" 7° dy =1+ea2*?T(p)—are [a eY(a+yy dy. r / y p—1 ) Thus F,, (x) — & x? T'(p)=— loo ey {(1 + 4 -—l1 f dy. “0 / y\P . : ¥ If now we expand i ) —1 in the form, valid when |Z\< il, = (Il) oe -(e—”) (y\” S aes: { =a nm} Pe (1) and substitute under the sign of integration, we shall obtain an asymptotic expansion of the integral when |x| is very large and |arg2|<7 since the expansion (1) is summable over the area of the plane dissected by a line passing from the point —z away from the origin to infinity*. Finally therefore we have the asymptotic expansion (A) of the previous paragraph, if R(p)>0. When R(p)<0, and p+n(n=0,1,...0%0) is not zero, we can at once establish the same result by substituting contour integrals for the line integrals which have been employed. * See the general theorem proved in § 5 of my original memoir. Vout. XX. No. XI. 36 270 Mr BARNES, ON FUNCTIONS DEFINED BY PART We The function oF, {p; x}. § 20. We now proceed to consider the function 2 Pal a ae, a =a i {p; x} aa Wr Ems) I (p) ph Se Ged ue. ae T(n+1)U (p +n)’ wherein p may have any value real or complex which is not zero or a negative integer. The function satisfies the differential equation dy dy A — —y=0. dae Pda 4 A second independent solution is a'°,F,{2—p; a2}. The differential equation has been considered by Pochhammer *. The reader may feel surprise that the equation was not considered before the dis- cussion just completed. The reason for this, however, is that by a relation due originally to Kummer+ it is possible to make the theory of the function ,f,{p; z} a particular case of that of the function ,F,{a; p; 2}. And this method of treatment seems to be more simple than any other which we could have adopted. The reader will notice the connection between ,F,{p; 2} and Bessel’s function. i (= ae 5 ey ———— = 2 Taal awl)! and therefore a z\-” es a) ( ny o fre ; el (5) I= 3 poop pa ery eth a All the following investigation can therefore be translated into an investigation of the nature of the asymptotic expansions of Bessel’s function J,,(z), when |z| is large, for general complex values of the parameter n, negative integral values being excluded. § 21. We proceed now to prove Kummer’s result oF, {o; a} =e F, fp —4; 29-1; 403} =e" F, {fp —4; 20-1; — 4a}. m pg yore 2 fa) ent} an We have é ll {p; x} ital 7 RETO REET Now consider - es P(2n—s) (4a)? ds aT7lLJ ¢C taken round a contour C' enclosing the points s=2n, 2n+1, 2n+2... as in the figure. . Se 0 an 3n+1 * Pochhammer, Mathematische Annalen, Vol. xxxvur1, pp. 228—240 and Vol. xu. pp. 174—178. + Kummer, Crelle, Bd. xv. p. 139. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 271 By Cauchy’s theory of residues it is equal to 2 ( " (427) = 1 So See = ee »\n aR r! r : Ce) ; eas T'(p) @ (2n — s) (4x)8" H e agl cE f : hee > ence 11P3 2 Dirt | eee Now by the multiplication theorem for the gamma function 2i—s—1 2° / TI (2n —s) = — —T(n-5)r (n-5 +5). V0 \ 2, Bier 12 s\ Sed C(o) 3 fo P(n-5)P (2-545) : Qa n=0!c2NV 7 T(n+1)0(p +n) , and this integral may by a similar argument to that employed in § 14 be written, if R(p+s)> 1/2, oxl/2 Hence ee Fi {o; «} =— = i & s ee allie las! ips be bok Vite 4, we have eo? BF, (oe arile Faye yp Zee * T(p) (pep +s ~4) G. if R(p+s)>4 by the multiplication formula. 2me [(2p+s—1) V7 — S (et a D2p+2nv—2 P(p)U(p+m—+) mao m! var T'(2p + m—1) age Ga tet) ae (ip) Fy T(m+1)° (2p +m—1) Vr =F, {p—4; 2p—1; — 4ar*}. We have thus shewn that oF (0; a} =e?" {p—4; 2p—1; — 4a}, provided }-— R(p) a}. For by the previous result (§ 16) we have FF, {op —1/2; 20 —1; 4%} aqil2 TP (2p—1) I'(p — 1/2) I (2p — 1) I(p —1/2) (40%?) -072 FP, {fp — 1/2, 3/2 —p; 1/(4a°*)} (— 4017)" Fy {p — 1/2, 3/2 — p; —1/(4a**)}. This result is valid if argw|< 2m: we then have jarge’*|<7. If arga=@ and 0<|0@|<27, we have arg(«'*)=6/2, and we must take such a value of (4x”)!2-? as is equal to 2} xe?, where arg (a?) = (1/4 — p/2) 0. We also have arg (— «'*)= 0/2 —7 if 6/2 be positive, = 0/2+7 if 6/2 be negative ; and therefore (— 4a}??? = 21-2 gpl/s—p 2, where arg (a4?) = (0 — 2a) (1/4 — p/2) if 6/2 be positive, = (6 + 27) (1/4 — p/2) if 0/2 be negative. Thus, since I (2p —1)=V (p — 1/2) I'(p) 2% 2/7, i{p; «} =a on we [e"” F, (p—1/2, 3/2—p; 1/(4a**)} + Pe" FP, (p — 1/2, 3/2—p; 1/(—4a*)}], where P = e="">), the + or — sign being taken according as 0/2 is negative or positive, and where a!4~e?= (a1?)2-p and | arg a!?| < 7 *, We see that the complete asymptotic expansion for ,F,{p; «} persists in form as x makes a complete circuit round the origin, except when « crosses the positive direction of the real axis, when there is a sudden change in value in the second (and then negligible) series. § 24. Consider now the asymptotic expansion of J,(z) when |z| is large, and n has any finite value real or complex except negative integral values. The quantity n may be zero. 2 We have IMA) = 6) rom ff bn Soil eee Dy TGS) > || "ee * This result is due to Stokes, and appears substan- Cambridge Philosophical Transactions, Vol. x1. (1868), pp. tially in his long-forgotten paper ‘On the discontinuity of 414—418. arbitrary constants which appear in divergent developments,’ 274 Mr BARNES, ON FUNCTIONS DEFINED BY We assume as part of the definition* of J,,(z) that when n is complex 2” = exp {n log 2} and that the logarithm has its principal value whose argument lies between — 7 and 7. Putting 2a! =.2 in the formula of § 21 we have Z\” 1 Py ea ” 1.9 - Qe}. Jn (2) (5) T@isey ha ee Therefore for all values of argz we have asymptotically 5 (ee POR I a ey ee | Int) =(5) Pint 1) P(n+s) [ oe Ent | ve ig} 1 1 + e-#(— 2uz)"* Ff, \" +4, 4—-—n; —-:; \] ; 1 202) 2” z —n—-} : 1 or IA) = Jas G (Qype lie \n +4,4-n: sal —1z -\)—n-4 i Te edie te 1 \ +e" (— uz)" * Ff, n+ t—n; aoe) : (A) Now the argument of (iz)-"~! lies between +(n+4)m and therefore we may put (Zz) = es (+3) g—n—i 2 ’ where z-”-? is made uniform by a cross-cut along the positive half of the imaginary axis, and has an argument which lies between $ar(n +4) and —$7(n +5). Similarly the argument of (—1z)-"~? lies between +(n+4)7, and we may put (= re) = ez nth) zn} where z-”-} is made uniform by a cross-cut along the negative half of the imaginary ; ; : oT ‘ 7 axis, and has an argument which lies between —~g +3) and 5 (n+ 4): 5 . . . T a If now 2 has its principal value whose argument lies between + =? and with the previous convention for the factor 2” which intervenes in the definition of J,(z) we have, putting 1. ‘When R(z) >0, a (z) = 5 A | oto P + ent se (nth) Q| : 2. When R(z) <0 and J(z)>0, 1 ete P + ent = (ath) Q| = Z , Vm 2 3. When R(z)<0 and J(z)<0, df. (z)=- vee jer Few jpae ene nth) @| : Jn, (Zz) = These are the complete asymptotic expansions for J): * A different convention is adopted by Nielsen, Cylinderfunktionen, p. 19. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 275 If we ignore series which are infinitely small compared with those retained we have: 1. When J (z)>0, 1 ur 1 Jf (Z) = ——___ gt g (n+) Q. V2 23 2. When J(z) <0, 1 or 5 Jn @) = —— 65a (nth) P. V 2ar 23 When A(z)>0 the dominant term in the expansion of J,(z) may be written 1 7 SSS ES Sg SF | NV Qar 23 par a More completely we have, when R(z)>0, > Jn (2) a = | ens Je-Fo+y| aa z sin [2-Fo+y]t, nat = ree = ll em ya < i D(n +2m +4) P(—n+ 2m + $) sb gA fi n=o (nm +4) 0 (—n +4) TP Qm4 1) Quzy" 2 4 jie ik T (n+ 2m + 8)T(—n+2m+ 8) Ji Sy ING Ose $)P(—n+4) T(2m + 2) (2Qezjpm * 2° The quantities Z; and J, are such that their moduli tend to zero as |z| tends to infinity, for any finite value of hk. This is substantially Hankel’s result. When z is not real, the complete series, of course, is not such that the modulus of the difference between J,(z) and the first & terms . ian 1 : : of the series is of order less than — when |z/ is large*. be § 25. We will next consider the nature of the asymptotic value of ,¥,{p; «} when |#| is large and | arg#|< 7, as p tends to positive infinity. We have Ki {p; a} =e" F, {p—1/2; 29-1; — 40}. If now R(x?) >0, we have (§ 7) FP, jo — 1/2; 2p —1; — 40%} Lai T(2ep—1) V(p—1/2+s) ; =-—>— —s) 3-5 4a!) ds ; 1 nt ) (51a Geta), 0” (1) and we may take the contour of the integral along a parallel to the imaginary axis through the point s=1—m, if R(p) be large positive and Sm, m being finite real and positive. Let us now consider what happens to the integral when R(p) tends to positive infinity. * With this series the reader may compare Forsyth, funktionen, pp. 154—156, &e. The history of the investi- Linear Differential Equations [Theory of Differential Equa- gation has been considered in the introduction to my funda- tions, Part m1. Vol. 1v.] p. 331, and also Nielsen, Cylinder- mental memoir. 276 Mr BARNES, ON FUNCTIONS DEFINED BY As regards that part of the integral for which |s |<, where ZL is a positive quantity as large as we please but such that Z/p is small, it is evident that T(2e—1) T(p—1/2+s) 1 P(2e—1+s) VP(p—1/2) ~~ 2*° When |s|>Z, where L/p is small, we may put |J(s)| =p, and on the corresponding parts of the integral w will take all values from e(>0) to ». And now, if R(s)=u, we have, when p is large, DRGs) Liga 2 cs) _T(2p—-1)T [p(t m) + u—1/2) T(2o—1+s) T(p—1/2) T(p—1/2) UT {p(2+m)+u-1} (2p)??-*" e—P [p(1 + ope) Pp Owe ea (tue) er Pie “pee” [p (2 + ap) Chu H=88 oe Eu ” approximately, the upper or lower signs being taken together. D (2p — 1) P (p Es 1/24 8) — 92p—3/2 pp—i2 [p v1 its fr mee I'\(2p —1 + s) I'(p — 1/2) x [p / 4, ae payete—se ea tan! /2 = 9-38 (les peyee (4 + ie) ee Suppose now that te: y =p tan w/2 — tan p+ log V1 + p?— 2 log V1 + w2/4, dy p/2 a bh b/2 Ther a= = a sion du 1+ p?/4 ee pp hee a 1+ p?/4 = — tant yw/(24+ 1°). Hence exp {p [w tan w/2 — ptanp + log V1 4 pw? — 2 log v1 + w’/4]}. + tan w/2 — tan w Hence dy/dw is always negative and therefore y constantly decreases as increases. Now, when »=0, y=0. Hence y is always negative if ~ > 0. We thus see that, if m be finite and if | arga'?|< 7/2, that part of the integral (1) for which |s|>Z, where LZ is as large as we please but such that Z/p is small, tends to zero as p tends to infinity. We now have, if | arg a | <7, Fo; a} =e? I, (#) aim) 29 where J,(#) tends to a definite finite limit as R(p) tends to positive infinity for any finite value of |#| however large, and where |J,(#)| can be made for all values of p (such that R(p)>m) as small as we please by taking | «| sufficiently large. The first part of this theorem can, of course, be stated in the simple form that, if | «| be finite, Lt .f,{p; «}=1. pHa § 26. We have seen that ,f,{p; 2} and a'°,F,{2—p, w} are solutions of the differ- ential equation Equally each of the asymptotic series which occurs in the complete asymptotic expansion of ,F,{p; 2} is a solution in the neighbourhood of infinity. It is natural then to seek a combination of the two functions which shall be completely represented by one of the asymptotic series. SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 277 We will prove that asymptotically, if | arga|< 7, = oe —p = a Tgp s— 2 PQ =p) 0F.(o3 a} tal (ps 2—p; al =Vew * Fp; pd; ah. In the first place we may shew that the combination on the left-hand side is, when |a| is large and |arga#|<7, smaller than any positive power of 1/ «|. For consider the integral = i T(—s)Q — p—s) «ds, round a contour which embraces the positive half of the real axis and encloses all the poles of the subject of integration. It is evidently equal to = 2) eae 2) + 3(- jd se 2) n= =0 T (n+ 1) n=0 ra (n+1) This in turn may be written = wa” = Tayler a T@+l)l(p+n)sin ap = SCE INCE pt+n)sin(p—1)7 =TQ—p) Fi(p; 2) +2 °T(p—1) Ff, (2—p; 2). Now, if |arg2|<7, the integral vanishes round that part of a great circle at infinity for which R(s)>—k where & is a finite positive quantity. Hence when || is large it is smaller in value than any positive power of iol" L | To prove the asymptotic expansion given we use Kummer’s result. We have G21) '(1—p)oF,(p; 2) +a°*P(p—1).F,(2-p; 2) =e" (T(1—p).Fi(p—4; 2p—1; 4a”) + 2° T (p—1).F,(8—p; 3—2p; 4ax1”)}, and this expression is by § 9 asymptotically equal to _a([Ed—p)PQp—1), ae eTe—Il (3 —2p) ; ‘ 1 om lp peael? Tene aap | P(e—b $75 ge) wae fl —p)PQp-1) , P(B—2)T(p— 1a al ae Rucernccos: + TG =p) (4a Fite, p-4s gaat Now by the multiplication formula for the gamma function T (22) _ 2"> hence it may be readily seen that P(1—p)F(2p-1) ,TR-%»)T(p-))_, T(p—3) Trg-pae ae P(~p)P 2-1), Pe-)@~2)_ Pe-)C He * Te=—/) CH remembering the prescription which we must give to the functions (— 4)°-? and (— 4)**-», Hence finally we obtain the given result. . Vou, XX. No. XI. 37 278 Mr BARNES, ON FUNCTIONS DEFINED BY § 27. We proceed now to give a direct proof of the previous theorem and to shew that, if | arg |< 3z, Lh 1 T(1—p) Fi {o; 2} +l (p—1) oF [2—p; al =Vre * 2" .F(8—p; p—4; —1/(40%)}. We thus extend the range of validity of the asymptotic equality from | arg 2|<7 to jarg a| < 37. Let S(s) denote the function defined when R(2s+p)> 1/2 by the series = T(—s+t/2) Td—p—s+t/2) wo | 2+ 1)PEC/24+1/2) We will first shew that S(s)=— RTP ort TP (— 28) P'(2 — 2p — 28) T (28 + p — 1/2). Tv If R(2s+p)>1/2, we have & RiGee +) Gps Ds s PG s+t+ 1/2) P(3/2—p—s +8) US) =. 2. mecca ECan Ges T (t+ 3/2)T (+1) =0 IRC) V0 Ce =,F, {= 8 1—p—s: 1/2; 1) ( Fase ) + F, {—s+1/2, 3/2—p—s; 3/2; 1 By Gauss’ theorem we therefore have sf PEOrdse=s). uy, Dente nGs epee S@=Te+2-1) pasts 12)* Ta+arers SFG pee ep 2) =['(2s+p 1/2) Dajo+a)T (+8 — 1/2) = — TPP 2s) (— 9 P/2—p—8) PA -p 8) PAs +p 1/2) {1 + tan vrs tan 7 (p+ s)} = —SE7P geet TP (— 28) T' (2 — 2p — 28) I (2s + p— 1/2). Let now CO be a contour embracing the positive half of the real axis and enclosing the points (es PA boon Etol Wes), P=} B= fy, Sase Then by Cauchy’s theory of residues for all values of | arg) we have | ne | T(-)T 0 =p—s) ads C Qarvs ¢ =T(1—p) oF, {p; #} +a P (p—1) oF, {2—p; a}. Also we have oall2 ol =a T(—s) (1 —p-—s) 2¢ax+t# SiN is ae, t=0J ¢ mea : ames And hence by the methods employed either in § 6 or § 14 we have e 1 . olla = Y 8 lo ee I |r = [s (s) ads, ; é 1 if the contour C excludes the points i-$-> R= OMe eo ets SIMPLE TYPES OF HYPERGEOMETRIC SERIES. 279 We therefore have e? T]p/m = 2» STP - | T (—2s) I’ (2 — 2p — 28) T (2s + p — 3) 2" ads, Cc 2are Now if |arg2|<3z the integral will vanish when taken along a contour at infinity for which R(s) is greater than a finite negative quantity. It is therefore equal to minus the sum of the residues of the subject of integration at the points 1/4— p/2—n/2, n=0, 1, 2, ..., & together with J;, where J; denotes an integral along a contour parallel to the imaginary axis which passes between the points s=1/4—p/2 —k/2 and s=— 3/4 — p/2 — k/2. s 1-2p 5 = 5) Q/< ol? _ _cosmp —~ & Tint+p—1/2) (n+ 3/2 — p) Thus ¢ I] “s 7 ¥ n=O (= 4gpV2)n 3 Je 1-2p =a 4 .F,{3/2—p, p—1/2; 1/(—42)jin + Jk, where | J; 24-e?-*?| tends to zero as || tends to infinity. We thus have the result stated. § 28. This result, since it is valid when | arga# <37, is equivalent to three different results. In fact, if —7 al ie) FY fet Sire ME my PARTE OS ae Se Te pNadiy deo orb Wi yong” teehee are NRA escruin Osh Lets luted } ertin nat L uy tiayae ue re i? é q¥ va rrpives ’ i t@ i’ . ‘ f ru ‘ ' rite, i i) pee ia i ‘ : (A) ’ , iig 7. on \ ies ‘ i i! > 1 joka NE . Thay u ‘ 5 — a i os iy lets sight Add mitt: cule Vey, Hanae gy ee j we ] al a ‘ “iat « 4 ¢ ? ‘ { i% if ie in 34 § if J 4 ! > - 7 cay j in j esini ' * Prt . ‘ \ t Lf ie i Te A } ' hae " iy) Py * be ~ NAL ele ar j a a, wad olay des Aah a. eer yee bad 7 ( * uk s terre] aT ae stlictt 2 itn date at On) SS teri wr 605 a aal vl vt y pals hh Ta ee Ss » pled | fos ' y = > et) or - — ; —_ _ Wy i. cae ns 7 XII. The application of integral equations to the determination of expansions in series of oscillating functions. By H. Bateman, B.A., Fellow of Trinity College. [Received Dec. 24, 1906. Read Jan. 28, 1907.] § 1. THE theory of integral equations as developed by Fredholm* and Hilbert+ has provided us with a new starting-point for a systematic study of the important classes of expansions which occur frequently in the mathematical theory of vibrating systems. The analytical result upon which the present theory depends is that the homogeneous integral equation b an ee i, eee ty Ga (1), in which /(s,¢) is continuous for values of s and ¢ on the path of integration, only possesses a solution ¥,(¢) different from zero when the quantity 2X, is a root of a certain transcendental? equation 6(A)=0, and that corresponding to each value of X and each function 1, there is another function ¢ for which the adjoint equation b bn()= An | TEUGE EY PRE NRE ce 58 A os eee te (2) is satisfied. If the equation (1) is multiplied by the function ¢@,,(s) corresponding to a different root Am and integrated between a and b we have b 6 fb [i ms) yn(s)ds =u ff bm(s) (st) Y(t dst rb =>" | bm (t) a(t) at, b which gives | CADP (OGe =O, (Mag Ika) asonooscneqonconase.ocesno6ac (3). a * Acta Math. xxvu. (1903). + 6(A) is in general a transcendental integral function + Gétt. Nachr. Hefte 1 and 3 (1904). of \, but it may happen that in some special cases it is a rational integral function of A. Vou. XX. No. XII. 38 282 Mr BATEMAN, THE APPLICATION OF INTEGRAL EQUATIONS TO THE If 2, is a simple root of the equation 6(A)=0 there is only one pair of corre- sponding functions ¢$,(s), Wn(s), and the arbitrary constants multiplying them are chosen so that | GNU GE ee (4). When however XA, is a multiple root there may be several different pairs, but Fredholm has shown that we can find a set of linearly independent functions yy,” and a corre- sponding set of linearly independent functions ¢,” so that b EXGiay is $n! (8) Yur (s) ds — é 4 Lig oom, Ace ae ee (5). The number of linearly independent functions w,?(s) cannot be greater than the multiplicity of the root A,: it is usually equal to it, but there is at present no evidence to show that this is always the case. The formulae (3), (4) and (5) enable us to determine the coefficients in the two types of expansions F(s) = = OnWn (s) 2 he adcnest etait he (6), AB)= >= OOO ) whenever uniformly convergent expansions of this kind are known to exist and sometimes when the expansions are non-uniformly convergent. The above analysis however does not indicate how the adjoint set of functions ¢,(s) can be determined when the functions v,(s) are given or vice-versa. The object of this paper is to extend the analysis so that this defect may be remedied. Starting from a single integral equation I (s) = ba(s) =r [kes iby (EO). dbancncscearssvcetoses-scecteer eee (7), whose solution ¢,(s) 1s known, we can deduce the law of the coefficients in the expansion of an arbitrary function in the form > Ay pa (s) SSSSSOOOOCOOOCOOOCOOOOOOOOOOOOOOOOCOOOnO Mo noror (8), where the quantities X are the roots of the equation in obtained by iniposing a linear condition of any kind upon the function d), (s). Since the functions f(s) and /(s,¢) are at our disposal it is evident that the results obtained in this paper are of a very general character. The integral equation due to Hilbert, which is chosen to illustrate the method, leads to some interesting results in connection with expansions in trigonometrical series and puts in evidence the power of the new analysis. § 2. Many of the well known expansions in series of oscillating functions connect themselves naturally with the problem of determining the values of » for which a linear differential equation of the form LUN = Ohnanrgaveor eset acis oteescasctesnreiceee tect eit (9), DETERMINATION OF EXPANSIONS IN SERIES OF OSCILLATING FUNCTIONS, 283 where Z is some differential operator, can possess a solution which satisfies certain linear conditions. It is natural then to expect that the problem of determining the values of » for which the solution of an integral equation may satisfy given linear conditions will also lead to important results. This problem really includes the previous one because a linear differential equation can in general be replaced by an integral equation. Let us consider then the integral equation f@)=4(8) =r fk, Oe ee ae (10), concerning ourselves with the solutions which satisfy various types of linear conditions. If X be regarded as the quantity at our disposal only one linear condition can be imposed and then the corresponding values of » will be in general the roots of a certain transcendental equation. If we impose the linear condition OENSE, (CSS hrtaasocasqnocagcecaooncaooscewecaoncer (11), supposing A, to be one of the possible values of X we shall have f(e)=c-r | TEC ee atn ates 2 ean (12), Combining this with the equation b F(s)=b1(8)—ro | els, 1) Go(0) dt, we obtain the homogeneous integral equation EO | Patan (ey aie CY A ue ee (13), where k, (s, t) = k(s, t) - 7 Ie (Gs. b) sive cases catlanesoesisas weiss ets (14). Now in order that this equation may be satisfied 4, must be a zero of the integral function 6,(A) connected with the equation ¥O)=xO-> | hE AOE ae eae ee Ck (15). Accordingly, we can anticipate that the function 6,(A) contains ¢(#)—c as a factor, for 6,(X) is zero whenever 2 satisfies the equation ¢(«) =c. The solution of equation (10) is known to be given uniquely by the formula* Re) = FG) om I TAGE GN GE ost s.secieee (16), except when 2 is a zero of a certain integral function 6(X) connected with this equation. * The formulae quoted will all be found in Fredholm’s paper. 38—2 284 Mr BATEMAN, THE APPLICATION OF INTEGRAL EQUATIONS TO THE The solving function K (s, t) depends of course upon 2» and is infinite at each zero of 5(A); moreover in the neighbourhood of a simple zero , it has the form ¥n (8) bn (t) aan ray 55 F(s, t) Cece cece ee eeeeceseeecvssssessseseceas (17), where F'(s, f) remains finite and y,(s), $,(t) are the functions mentioned at the beginning of § 1. It will be verified in § 3 that the corresponding functions 6,(A) and K,(s, t) for equation (15) are given by BA == "30 LGB) = LAG, )-o9— K (a, t) Now it follows from equation (13) that when A is a root of the equation $(«)=c, the corresponding function ¢,(s) is a fundamental function for equation (15). To find the adjoint function we apply formula (17) to the function 4X, (s, t), remembering that Wn(s) corresponds to ¢,(s), and ¢,(t) to the function we want. Calling this function 0,(t) we have dy (8) Cat (A) — 2») Ky (s, #) =f) Ko(w, t) - $ (#) K, (@; t) = [kK (a, t)la=ay> the terms K (s, t), #(s) and K (a, t) remaining finite for 7=A,. The function 6,(t) is thus given by the formula A, (t) = =e" and so if the function K (#,t) is known the coefficients in the expansion of an arbitrary function #'(s) in the form 1W@Qy= = "AN (G) Di Aoy (8) ics esege sc omece iat sces oeseeeneneeen (20), for the different values of X, which satisfy the equation CY OSA ==) onanc oggs00n oc HOOD UapuonSBHECaneoISa30005¢ (21), are given by FAN (NS) = [’ ee o(, t) SEG). GEM sasas'silacs cs erleseielseceoen tenet (22). : dr, $o(@) It will be noticed that ,(s,t) is obtained from k,(s, t) by replacing the functions jf and k wherever they occur by ¢ and K respectively: this rule enables us to write DETERMINATION OF EXPANSIONS IN SERIES OF OSCILLATING FUNCTIONS. 285 down the functions 6, (A), K,(s, ¢) for the integral equation obtained by imposing any other linear condition upon the function ¢. For instance if we require $(s) to satisfy the linear condition [ems i ile PR ee Bees (23), the values of A, will be such that b b pb | p(s) f(s)ds=c—X, | / k (s, t) p(s) $)(t) dsdt, b F(s)= br(s)—o | (s, 8) do (t) a and the equation which replaces (13) is 6 bi(s) =r | ils, do at - f(s) | : p (2) le (a, t) dat ........cee00- (24). c— [ p(x) f(x) da~ * “a where k, (s, t) = k(s, t) + It will be shown in § 3 that for this equation _ p(s) me K,(s, t) = K (s, t) + —*" —__|_p(e) K(@, t) da, c— | pla) bla) de" rb c— I p(x) (x) dx 8, (A) = c—] pla) f(w) de 6 (A) b | p(x) K, (2, t) dx 6, () = a, |, PC) $(a) de As an example of the application of these results we may consider Hilbert’s integral equation 1 b)=S)-r] ko D SMa, : : , _(l=s)t t = en i>4| 286 Mr BATEMAN, THE APPLICATION OF INTEGRAL EQUATIONS TO THE and if we take the particular function f(s)=s, the formula 1 $(s) =f (8) +% i K (s, t) $(t) dt /0 gives (Gy DY 8 te ee eee (27). The previous work thus enables us to determine the coefficients in the expansion of an arbitrary function in the form TOK GI isl B77) dcapnencedecodbocdo anoanonasondaceno0" (28), A where the summation extends over the values of for which the function ¢(s) satisfies any given linear condition. Suppose in the first place that the condition is ¢(x)=c and write A,=m?, then the values of m must satisfy the equation SIN (neat omponedeaaaeosaueandoensconcDb050050600%° (29). L - [a cos aVA, sin VA, — sin 2V Aj cos Vv] d New. rg (a) 2VX, sin? VA, @ COS mx — C COS™m 2m sin nv Calling the different roots of equation (29) m, m,...m,... the coefficients in the expansion F(s) = By, sin ms, as determined by formula (22), are given by oF sin m.[x cos mx — ¢ cos m] 7 1 = sinm(1—2) i F(t) sin mt dt + sin max i F(t)sinm(1—#)dt...... (30). 0 x 1 Again, if the condition is | $(s)ds=c, the values of m are given by 0 eis sin ms | : Ib= eG, 9 sinm m tan 2 1.e. m 2 sin en Si ee dest) 1 In this case I K (s, t) ds = 0 DETERMINATION OF EXPANSIONS IN SERIES OF OSCILLATING FUNCTIONS. 287 d = 2 — ————— ee a) GNe. l" 0 eG) de ot 32 2! pa Seedy 2 1 = vr%_ 1 VA 1 ( m ), and so the coefficients in the expansion of an arbitrary function in the form > B,, sin ms, m where the summation extends over the values of m for which m tan > =em, = are given by 4B,, sin - eae — 2c cos? 3)= ie F(t) sin — be .sin oo Sy) CEES near eee (32). § 3. Verification of the formulae given in the previous section. In this section we require a convenient notation for the result of imposing an arbitrary linear condition upon a function ¢(s). The linear conditions which we shall consider will in general be of the form “a ie a(s) &(s) ds + > = Pn (y) - C= =/CONShANU ances eeten eens (33), and it will be supposed that the operator which produces the left-hand side when it acts on $(s) is such that it can be introduced under the integral sign whenever ¢(s) has the form [eo t) y(t) dt. If the result of this operation is expressed by replacing the variable s in the function ¢$(s) by means of a star, the effect of imposing the given linear condition upon ¢(s) will be represented by and the effect of our operation upon the function represented by the above integral will be b [ koe Ox Oat Let 2, be a value of > for which the function ¢$(s, 2) given by the equation F)=4@)—r[ ko, DSO at satisfies the linear condition ¢$(*)=c; then we have F#)= c= de f RC, COL eae en (35). This relation may be used to reduce the equation F(8)= $00) = | bOs, #) (6) a 288 Mr BATEMAN, THE APPLICATION OF INTEGRAL EQUATIONS TO THE to the homogeneous form eT te ye ae $y (8) = Ao Hi E (s, t) F(®) = of (*, 6 | Dy (G) Otemectee caceews anacsee (36). Now according to the rule stated in § 2 the solving function of the integral equation b (5) =X) =D [Ba (8, 1) (0) do oerrerecereeensesennennnnnnn (37), in which k, (8, t) =k(s, t) Sar k(*, t), should be given by K, (st) = K(s, ¢) = ae KG (8, ty ive cces see oeetesee oaaenes (38). To verify that this is the case it is simply necessary to show that the characteristic relation b ky (s, t) = Ey(s, )— [by (6) Ki (@, 8) do -oescesessseeessneeen (39), which is obtained by substituting in (37) the value of x given by x=¥O+r | KG, t) y(t) dt, is satisfied. Now r IE k(s, z) K (a, t) dx = K (s, t) —k(s, t), b - X [ k(*, x) p(x) dx = $(*) —f(*), db S| k(x, a) Kile. do Kiet) eee 6 » [ (3, 2) 6 (a) de = $(8) Fs); accordingly, when the values of k,(s, 2) and K,(a, t) are substituted, we have IN if: k, (s, x) K,(@, t)dx = K (s, t) — k(s, t) — ee [(s)— fF (s)] FS) ry f(s) K(%, t) Saye [AG )—k(*, Ol sa =cl1e@) =o [o(*) — f(*)], K(x, k(*, 2) , Se ae Pee $(s) + ee 2 F(, = K,(s, t)—k,(s, t); hence the relation is satisfied. 289 DETERMINATION OF EXPANSIONS IN SERIES OF OSCILLATING FUNCTIONS. To obtain the value of 6,(A) we make use of the formula u d [ Ke.) dt=— 5 Dog 8.00) (40). rb | K(x, tbat Substituting the value of K,(t, t) we have d d 1 == [log 6, (A)]=—- ar [log 6 (A)] — aS. “b To obtain the value of | K/(s, t)¢(t)dt we multiply the equation b k(s, t)=K(s, #)-—» | k(s, 2) K (a, t) dx by $(t) dt and integrate, thus “b rb rb “) | ko de@d=| Ko HeOd-r| kG, ade] K(w, 1) oat But if we differentiate the equation “bh f(s)=$(s)—rX| hls, 2) b(a) de, with regard to A, we obtain “b b 0= FO _ Pas, a) G(a)dv—r [ k(s,0) Pa Comparing this with the previous equation we see that rb F) | Ke ne@dt=~ do) Pere ik sv eee (41), = PA rasedeecsnisxene ceasvdseegseseceae (42) [K(*, )e@a= and therefore Substituting this in equation (40) we get — © fog 8 ay) =— & flog ay] — © flog fg (#) — 6] ae ay + °S TR ee J o(*)-e¢ ¢ d Now when A=0, 6 (A)=6(A)=1, $(s)=/(s); therefore on integrating the above equation we get which is the other formula that was used in § 2. To determine the function 6,(¢) adjoint to ¢,(s) we must find the limit of (X,—2) Ky (s, t) Substituting the value $ (s) K *, t), g(w)y—0 9 39 when X=Aj. LAG N= IG) = Vout. XX. No. XII. 290 Mr BATEMAN, THE APPLICATION OF INTEGRAL EQUATIONS. and remembering that K (s, t), $(s) and K(x, t) remain finite for X=, we have Tit (hg Ge) — bu(s) Ko (% t) A=Ao d aX p (*) accordingly @,(t) = Ki(#, 0) EEE Se onscknnonomeaancoay (44). ’ a Bn an, ? 5 ee d Also since i K(s, t) 6(t) dt= aD. $(s), it is evident that the equation b | 0, (t) dy (t) dt => 1 a is satisfied. XIII. The Variation of the Absorption Bands of a Crystal in a Magnetic Field. By W. M. Paces, B.A., King’s College. [Communicated by Mr H. W. Richmond.] [Received 30 March, 1907.] Iy March and April, 1906, M. Jean Becquerel published an account of some obser- vations on the effect of a magnetic field on the absorption bands of xenotime, a uniaxal substance giving sharp bands. Two regions, one in the green, and one in the red, were observed, and it was found that a diffusion and doubling of the bands occurred, analogous to the Zéeman Effect for metallic vapours. The first series of observations* deals with light rays perpendicular to the magnetic field, and the variations of the bands are given when the optic axis of the crystal is (1) im the direction of the light rays, (2) perpendicular to the light rays and to the field, (3) parallel to the field. The most important results obtained were (1) Certain bands are displaced to a greater extent than would be expected from a knowledge of the Zéeman Effect in metallic vapours. (2) The bands of the ordinary spectrum behave very differently for different directions of the axis of the crystal, although the orientation of the ordinary vibration remains the same with respect to the magnetic field. Thus when the axis is normal to the field, and the vibration is also normal to the field, the green band No. 18 in Becquerel’s Table (wave-length 52214 up) only gives one component, slightly displaced towards the violet; but when the axis is parallel to the field, the same band, for the same direction of the vibrations, doubles, and the separation of the components is 0°53 wu in a field of 31,800 c.G.s. units. * Jean Becauerel, C. R. March 26, 1906. Von. XX. No: XMM, 40 292 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS On the other hand, the green band No. 22 (52457 uz) undergoes a large displacement in the first case, and hardly varies in the second. Again, there is a remarkable difference in intensity in the components of the doublets separated by the field. For example, in red band No.6 (653°71 wu), and green band No. 22, the component displaced towards the red can only be seen at the instant of the field establishing itself progressively. This component disappears as soon as the field becomes intense. In a later paper* the case of light rays parallel to the field is considered. If the optic axis is perpendicular to the field, the modifications of the bands for the ordinary vibration normal to the field and to the axis are identical with those given in the first paper. The same is true for the extraordinary vibration normal to the field. When the optic axis is parallel to the field the ordinary spectrum alone is visible. As soon as the magnetic field is established the edges of all the bands diffuse, and the bands double. The components are found to be circularly polarised in opposite senses. If, however, the axis of the crystal is inclined at a few degrees to the lines of force, it is found that for two symmetric positions A, 4, independent of the strength of the field, but depending on the thickness of the crystal, the components become plane polarised in two perpendicular directions, independent of the field strength, and turning through a right angle when the sense of the field is changed. A very slight displacement from these positions A, B makes this state of polarisation cease, and the bands again become circularly polarised. However, in the immediate neighbourhood of these positions a partial polarisation of the two components is observed. These phenomena are produced for all the bands at almost the same instant. In observations on the Zéeman Effect previously made it had always been found that the components of the doublets which exhibited mght-handed circular polarisation were displaced all in the same direction, and that the left-handed components were displaced in the other direction. M. Becquerel, however, finds that for a large number of bands this is not the case, but that the components are displaced in a sense opposite to that previously observed. We shall suppose the crystal to be made up of a series of electrons moving about centres under forces varying as the distance. For any one electron the intensity of this force along the direction of the optic axis will differ from its intensity in any direction perpendicular to the axis. In addition to these forces there will be forces of so-called ‘frictional’ type, due to the action of neighbouring electrons, causing absorption. We will first consider the case of a single electron moving under the action of the central forces and external magnetic field only. * Jean Beequerel, C. R. April 9, 1906. OF A CRYSTAL IN A MAGNETIC FIELD. 293 Suppose the magnetic field, H, to be parallel to the z axis. If the axis of the system is also along Oz, the equations of motion are mé =—kx + eyH, mij =— ky — eH, mz =— kz, e, m being the charge and mass of the electron, and hk, /, constants. If we suppose 2, y, 2 to vary as e”*, we obtain from these equations (— mp? +k) a, =cepHy,, (— mp? + k) yy = — cepHa,, (— mp? + k,) z, = 0. Hence the period of the z vibration is unaltered by the field. The periods of the w, y vibrations are given by (mp? — ky? = & H?p*, or mp? —k=+ eHp. If p, denote the value of p before the field is established, and p,+6p, the disturbed value, we have 2mp,dp) = + eHpy. Thus we now get the two periods eH el Po om? Pom," Hence an emission line of the system polarised in a plane perpendicular to the zy plane will be split up into two, owing to the action of the magnetic field, if the axis of the system is parallel to the field. The line polarised in the zy plane will be unaltered. If, however, the axis of the system is perpendicular to the field, and lies along Oy, the period of the z vibration is given by mp" =k, while the periods of the zy vibrations are given by (nip? — ib) | (ane Ihe =e? anc no useniisce sc ocoaeessaseetecests (1). In the absence of the field, the values of p are given by mpi=—k, or mpe—k,. The disturbance in the first period is found from (1) by substituting p=p.+6p, when mp;?=k. Provided that &—, is not small, we thus get 2mp, dp, (k — k,) = &H*p,’, &H’p, or op) = me ED 40—2 294 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS Thus the band corresponding to this value of p, is not doubled but is displaced by an amount depending on H® A similar statement holds for the band corresponding to the other value of py. Hence we get a doubling effect due to the field when the plane containing the directions of equal period is perpendicular to the field, but this doubling does not occur when the plane perpendicular to the field contains directions of unequal period. The above is to be regarded merely as an illustration giving us some idea of the nature of the effects that are to be expected in a crystalline system in a magnetic field. It can have no real validity since the ideal system contemplated is a powerful radiator, and its vibrations could not persist for more than an infinitesimal time. We now return to the general case. If £, 7, ¢ be the az, y, z components of the displacement of the electron from the position of equilibrium the equations of motion will be of the form 2 OE = Mm > =eX — 4cre? @ no 08 &—re ot’ with similar equations in 7, € Here e, m are the charge and mass of the electron; @ and r are positive constants; X is the # component of the external electric field. The last terms in this equation are respectively the # components of the central and ‘ frictional’ forces. The quantity ¢? is introduced into the coefficients of these terms to shew that the corresponding forces are independent of the sign of the charge. For a crystalline body @ will be supposed different in different directions; but we shall assume, for simplicity, that 7 is independent of direction, although this is in all probability not the case when the asymmetric structure of the crystal is taken into consideration, The components of electric current at any point will consist of two parts, viz. (considering the 2 component only) (1) The current of aethereal displacement produced by the electric force X, This 1 ox dor Ot (2) The current due to the motion of the charges carried by the electrons. This is equal to is equal to LeNV = , where NV denotes the number of electrons of any particular type in unit volume. Thus the components of electric current are given by AL OKO eeeae le aan +2eN a, 1 oY an We Fe ae + tN ae 1 0% 0g =-——+>eNV—. a aap on oN = OF A CRYSTAL IN A MAGNETIC FIELD. 295 Now suppose that a magnetic field H is acting. In consequence of this each electron : 1 : will be affected by a force 2 ele: A) where v is the velocity of the electron and ¢ the velocity of light in aether. Hence if @,, @:, @; denote the values of @ in the directions of Oz, Oy, Oz the equations of motion of the electron become OE 4ae 29& |e (On 0g Mm =e = eX — Oe cake a to (ae He— = Hy), On > Ane 0” 8 (06 0& mag =O gree + (Ge He 5 He), eC 4me, ee (0, am M wi = OL 0, gc re apie ag ae H,). Now let us suppose that light waves of period 27 are traversing the crystal. Then t in the steady state which supervenes &, », € will vary as er. Hence from the equations of motion we get Cy , Gi ef, — FF (nH — Hy) = =~ X, _ ee eS Oy enrs oor (CH,, fH.) = 7 K Cus Le etr,—c. A (GH — 12) = 74, Wy 6,H vhe Hl —— =, where 1 +6 cae Pee fa Pas rd, me I b = ete! Oe 1 dre?’ with similar meanings for A.p., ete. We have assumed the magnetic force in the light wave to have no appreciable effect. If this were not the case the equations of the field which we shall derive would no longer be linear as regards the electric and magnetic forces of the light wave. This would mean that the optical qualities of the body would depend on the intensity of the light. An effect of this kind, however, has never been observed *, Taking the magnetic field parallel to Oz the above equations become 0 e&X, — en = ee X, CNAs + LepoE = 2 Ve egr,; = 5 Z. * Drude, Optics, pp. 452 et seq. 296 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS Optic Axis normal to the Field. Suppose the axis to lie along Oy. We have G10, —Gesave 6,=0' say a Aree 70 i) Cee a? , 70’ Optic Axis N Ais Iie S \ s ES ee LE MS bie ta P Acre’ 6 = n=) Fie g dere? NaS ye ilety —— 6H bi = Bs ES Agere’ See i} Ha B= dorore’ 0 eEX — Leen = Ae XG U / , 0 enr +cepwE= 7, etr= a Zz. Solving for &, 7, € we get ArreE (AN’ — pp’) = NON + mY, Acren (AN — we’) = rO'Y — yp OX, Aretr=80.Z. Therefore the current components are given by Aerie = = (1 es — x) i = S aa ONX OX ONp’ ea Ob XN = pp’ sory = (1 +25), = dary y = (a += OF A CRYSTAL IN A MAGNETIC FIELD. ONN ONu If we put q=1+ Sue as ea WaT ONX yp’ e=1+ 25 , a cA 9 AX — bE AN = py ON = > = é,=1+ he ba ae La we obtain TYz = & at Wy, at’ 4 € i cs 0 meewatinn Pat Airy. = €; at Optic Axis parallel to the Field. The axis lies along Oz and we have 6,=06=0, 0=6. With the above meanings for X, X’, w, mw’ we get e&X — cepn = a XG (iene enr + cewE = re ¥; @' f) ae nO Vue Hence dary = Ot (1+2 ae a irs = a 4 = a (+s Gang ° oe Ne > f = Sy = tsa Writing qg=1+: eae? fied Nye jn bare &= - ne > ox oY we have 4772 = & OE + Wy, ra Ny eat ne Amy, = a é 298 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS Light Rays perpendicular to the magnetic field. If a, 8, y are the components of magnetic force in the light wave, Maxwell’s Equations may be written IL , cag Br V)=—curl (X, ¥, 2), 4Aar el (Yx» Yyr 2) = curl (4, B, 7). We shall consider light waves travelling along Oy, so that the quantities involved are functions of y only. Then Maxwell's Equations give B — 0, p= 0, laa_ 0 1 by_ aX OG ae cot oy a ne e 1 By’? ¢ a Dy. Hence, eliminating a, y we get Yy = 9, td 820 dw _ OX @ at? ay’ @ ob oy? Optic Axis perpendicular to the Field, and parallel to the Rays. Substituting in these equations the values of yz, yy, yz given on p. 297 we get e Y —w.X = 0, 1 ( ax et _ ex GN or Per) mee see AZ CO Oy? Eliminating Y between the first two equations, we obtain 1 ( Vyy\ OX OX alc =) a) ae als Cc €/ 0 dy? L t r = (t—py) = (t—py) t—pyy) Put X =Le , Yao M2 7=New =. LVo Then M=— Tf, €> VV 22 — bad po =«& , €s pyC? = €s. ads ON see : : : : Since «,=1+-—, and so is independent of H, it follows that the z vibrations are unaffected by the field. OF A CRYSTAL IN A MAGNETIC FIELD. 299 To find the effect of the field on the X and Y vibrations put pe=n(1 —tk), so that » is the index of refraction, and « the coefticient of absorption. Nay an ONp ON - Litas pe a aa 7 Then n2 (pS al See gg SE eS AN — we eas 6 Nx A! = pp’ from p. 297. The quantities A, X’ are complex, and it is due to the imaginary parts of these terms that we get an absorption «x. We have aie; ee Spas Rha chen a 7 Tt TT we pen & = dercte E darore , We shall suppose the frictional terms = = in X, X% to be negligible unless the impressed period 7 is almost coincident with a ‘natural’ period of vibration of the electron. Returning to the equation of motion of an ion given on p. 294 and supposing the frictional and impressed forces to be absent, we have Let = mae t &=0, ce b. or met bem e. Thus the ‘natural’ period of a vibration of the electron along the w or z axis is given by = w/b. Suppose that the impressed period is given by 7=Vb(1+9)=7,(1+g), where g is small. Then for ions of the type we are considering ee Vn ORE): =2g+e = neglecting g’, and so is small. , ee A neglecting g compared with the other terms. For electrons of other types we have i a7 Thy If >’ refer to these other types only, we have i 7 —_ — NN on (1-5) 6 ¢ = 5) (29+.2) = Vou. XX. No. XII. 4t Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS 300 Neglecting terms which depend on H* we get ON NV a x Qq+u bs di Again “12 is of the second order in H, and so will be neglected. 2 ‘ ,ON ON ‘Therefore pe=n(1—uyp=1+> 7. ke ae 2g +0. Ty Since — is real, it follows that the imaginary part of the right-hand side, on which « depends, will be a maximum when g=0. Thus the maximum of «, which gives the position of the absorption band, is unaltered to the first order by the presence of the magnetic field. Hence, to the first order, there is no displacement of the bands of the ordinary spectrum when the axis is perpendicular to the field. Optic Avis parallel to the Field. We substitute the values of yz, yy, yz given on p. 297 in the equations on p. 298. Supposing that = t—py “(t=py <@=p, Kale’ | Yo Me PN = Na ee we obtain by the same process as above M=— f, © a pe “| E2 e (4 S el ce: € x e = Pi- Since from p. 297 NG’ o Ss e=14+2 >? and so is independent of H, it follows that the z vibrations are unaltered by the field. Again, putting pe=n(1— cx) we get J SS SS 4 woke et | (722 = pw? ne (1 —ex)i= : Vnro as rae N@ NO > ee Dy z (142) ( +, a = ras 28 OF A CRYSTAL IN A MAGNETIC FIELD. 301 As before, we suppose the impressed period nearly to coincide with a ‘natural’ period, and put . r=Vb(1+9)=7,(14+ 9). Then for the particular electrons under consideration a V=29 +4.-, = 1 which is small, while for the other electrons and these are not small. The electric force in the incident wave may be resolved into two components, one along Oz, the other along Oz. It is the vibration corresponding to the latter with which we are dealing at the present moment. For the other electrons ~ is small compared with ». Hence for these, we have ay te 1 (1 os ~ 1 pees A+u Rr r ie b ( 1 ) He Thus if >’ refer to these other electrons only, we get yy NO yy NO_ 5 NOy Lee auth = =A-—A’, N@ where A=1+2’ 7 ) 1 ih ip KAS NOu | (1-2) TT: 7 T, \ A—A’+ me Hane UENO a a 2g+e. +p 2g+l.7— pe Thus n?(1—cxp= 41—2 302 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS : ; Lie: A Now dA’, » both contain H as a factor, and also peer a= will be of the same TG order as pu. Hence the terms in the numerator which contain A’ may both be neglected in comparison with the other terms. 9 g Ne (29 ++.°) Nes (29+ iS) we (20+ ye Therefore nm (1 —cex)?= A+ we ee NO (2g +4 “) A+ = a is 2 (29 +t “) =i ee i a eg as A} (29-412) — ut +00 (29-44. ) Now, when no magnetic field is present the dielectric constant ¢ is given by N aes 25h Se | where 7, gives the unpressed period, and 7 the natural period of an ion. We have supposed 7, to be not very close to 7, so that the frictional terms may be neglected. 2 Thus 1—— isa quantity which will never be very large for visible impressed periods, ans but may be fairly small. Hence, since ¢, does not vary very much with the period, and since there must be a large number of vibrating systems in the crystal (as is seen from the large number of bands in the spectrum) it is legitimate to assume that V@ is very small. Hence we may write u NO. A (2g +0.) \ gt et = J=+) right-hand side, determining «, will W(1—up=At+ Aj (2940. Hence if © is small, the imaginary part of the ai be a maximum when 49?— yw?=0, ie. when 2g=+ 4. These maxima of « determine the positions of the absorption bands. Thus, when the axis of the crystal is parallel to the field, a line corresponding to a vibration perpendicular to the axis is split up into two. OF A CRYSTAL IN A MAGNETIC FIELD. 3038 Remembering the result given on p. 300, it may be stated that, according to our present theory, a band of the ordimary spectrum remains unaltered when the axis of the erystal is parallel to the ray, but doubles when the axis is parallel to the field. The bands of the extraordinary spectrum remain unaltered in the latter case, while the bands of the ordinary spectrum corresponding to vibrations parallel to the field remain unaltered in the former case. Becquerel, however, found that a large number of bands in the ordinary spectrum varied more or less when the Optic Axis was parallel to the incident rays, and so the results which we have obtained above do not agree with observation, They are approximately verified for Green band No, 18 (522:14uu), which only varies very slightly when the axis is parallel to the rays. The other bands, however, in many cases double for this position of the axis, which is quite contrary to our results. Hence we must seek some extension of our theory to bring it into closer accord with facts. Becquerel suggests that an orientation of some of the molecules by the magnetic field might remove some of the discrepancies. We will therefore suppose the axes of one system of electrons to be deflected from the axis of y, and to take up a position in the plane yz. This position must be definite, otherwise the absorption bands obtained will be .blurred and indistinct. Consider a vibration parallel to Ox. We shall expect a doubling effect to occur, for we have shewn that when there is a magnetic force Ae perpendicular to the plane containing the directions of vi- brations of equal period of the electron, the absorption band is split up into two, When the axis of the electron system is deflected through an angle ¢, there is a magnetic field Hsin @ parallel to the axis, and so perpendicular to the plane containing the directions of equal period. Thus we should expect a doubling, with separation pro- portional to H sin ¢, to occur, We will see whether this will be the case, and whether a band corresponding to vibrations along Oz is similarly split up in consequence of this magnetic orientation. Orientation of the axes of one system of electrons by the Magnetic Field. We suppose the axis of the crystal to be perpendicular to the field and to lie along Oy. We also suppose that when the magnetic field is established the axes corre- sponding to one set of electrons are orientated and lie in the plane yz making an angle ¢ with the axis of y. 304 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS The equations of motion of these electrons will be altered. An electron of this system is subject to elastic forces of intensity represented by @ along Oz, @ along its axis, and @ in a direction perpendicular to Ow and this axis. Hence the components of the elastic force on this electron along the axes are 4reé aia along Oa, n (“3 cos? @ + a sin? 6) + €sin d cos d (3 - ) ” Oy, ¢ (> sin? 6 + — cos? 6) +7 sin cos d (4 is a » Oz. Thus the equations of motion for ions of this type are Msg = OM ag” Se ena On x (ane re Bs (“F ed =) ,0n eH 0& mot = ey —(“F ona - sin? b)1 sin } cos a r €— re Bf eee m oe = eZ — (= sin? fo ar cos? $) €—sin dcos (4. - a) n— Tre eS ot? (7 ot With the same meanings for a, a’, b, b’ as on p. 296 these equations become Ok 08 eH On ot ve Ie rect ear art 10°) = a _ on _ eH® 0& b af = Y — (cos: @ + § sin? ) y — (1 — 8) sin ¢ cos $f — a’ had meer ese 7 — (cos! 6 + 5 sin’) r-(; £2) sini ecninee dt? = are é One Ot’ where =. t If we suppose everything to vary as e7 and use the values of X, X’, w, mw’ on p. 296, we obtain 3 eEX — Leun = re Xx, : j Gin e en (A\’ + asin? d)—ea sin p cos $f + ewE= a WM, : 0 e€(X+a' sin® d) + eo’ sin ¢ cos $n = Z, 1 , where Oras cs he Solving these equations for & », € and writing, for brevity, p= — pe’ +rcsin® d + (v = ae ) a’ sin? bd + co’ sin? ¢, OF A CRYSTAL IN A MAGNETIC FIELD. 305 we obtain, for the components of electric current é OX {AN + (Ao +N’ +-c0")sin® } + iy’ Y (A+¢' sin® $)+ mwOZo sin gp cos =~ |X+>N. —_— a dary» = Es +N Me a sy —eOX (XA +0’ sin? d) + AO Y (A +’ sin? $) + APZo sin $ cos sary, ay [v+sy SSS SS aa - == hs fe 5 [4 +3N. wOXo’ sin pcos d — dO’ Yo’ sin ¢ cos $+ 0.2 (AN — ww’ +o sin? °| - — = —. =. Xp F . Of course in all terms under the summation denoted by ¥ which do not correspond to the ions which have had their axes rotated, we put ¢=0. y ‘ O'N Put et ae y, => — ane AX — he AN — BE 'N. = TONE e211 pee ee AN — Bh AN — he ON a= 1+ 2 Then we have fe INGLE NAS a Rie oY 4 SEY Pes ca sin? i) == —= -—— : sin? Tx = (6 Ves glee b)a te ; sin $) at +1 NEE in boos 5 IVGlo Ns) UeNEEoL oY NOwc rA+o . OX Amy, =| & — — .—-—,, sin? ——1(v,— —— . —;——, SIN” mY («. i dr’ , sl ) Tome (». 5 Stee ¢) ot _ Nee sing@cosd 0% p eta fa BE Ss \ a Nber ae ea Vis, OZ Nul00" 4 og god — NO o'sing cos aY ee %: pe r ee Ap stn aes p Ot Consider a wave travelling along the axis of y. If we put (X, ¥, D=Gh, W,, Pye” we obtain from Maxwell’s Equations as on p. 298, ( N@oup = dA+o° NGpeaeN eG Nyutc . 2 (a= ae a ai sin? ¢) M, +1(%— Fag eT sin ) N,+e. a sin ¢ cos dP, = M, p*e’, ? NGG Na Nat oN N@po Ato’ ., Néc sin ¢ cos } («- = DAN pp Sit) Ms — eC : pe ee ¢)ah+ ae p P= =!) Py+e. eve sin ¢ cos dM, — AG ¢ ae £080 NG eee 306 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS Eliminating J7,, V,, P, between these equations, we get for the period equation yr ‘ ane yy , N , N ' : («. = @Nxo sin $ ; a Mi ) («, = 6 Pp HAE F a 7 sin? db — pre) i p NA — py / \ p ny AA —pp ) 1 Bh Ng (. No" r i if in2 2, :) 5 - — 7-0" 3 r 5 in’ d — prc ; Né6c pe Ato . ) sin? & cos" ne = sin? d — 2c? SD Eee. (« p DAN = pp’ St ape N°*00' co" +5 p ~ (4 - 7S, Perea sin? 6) (», "PHS sas /sin® $) p AN — bye p AN = pb ’ be ‘ — N60" » he sin? 2 :) 3 ~ F sin? d — pc N*0a ee Ns x e.)| ="). Set a sin? ¢ cos? h {6 (x: ais | NED p? - <7 sin’? h |) + w Gz _ p AN — bE 6) Vibrations parallel to Oz. If there were no orientation the period of these vibrations would be given by Po C= €:. Suppose that when orientation occurs, the period is given by P= Pot Spo. Substituting this value in the period equation above, we get @Nrosin®?d Ato ONo pw Ato . ) Sin = eg Ee eee Fain hee One («. ;) (« €, 3 x Neue sin? d — 2p,c* dp, pe Nida waste eee i 7 as oa sin? @ + 2p,c7dpp Ta , , a N se sin® d cos’ d (« —€—-- a! a A Fil sin? d — 2pc*Bps) rar ’ TQ.' , +(,-"S Sie sin’ $) (v= EHS ES ie 3 Or ae sin? ¢) wee Ned * 3 =a 7 oe sin? d + 2p,c*dp, l'Oac0' . (i NO’ Ns: eke ; We cannot neglect the term 2p,c?dp, in the middle factor of the first line or the p “= pe , second factor of the second line, since e,—e, is of the second order in H. T\/ Tr Tr , ~ GNX sy ON _ 5 ON yn K or @—-eé& =a Dr’ = ep — Lt Ay N . Nae fr OF A CRYSTAL IN A MAGNETIC FIELD. 307 Vibrations parallel to Ox. When there is no orientation, the period of these is given by 2,2 VyVo Dear When orientation is present suppose the value of p to become P=Po+ Spo- Substituting this value in the period equation on p. 306, we get 6’ NXo sin? A+ 0° Be ONe ENE Go 2 («. eb $ . as 2 :) ( ae F aah sin? hd — 2p.0*8pa) p AN — whee \ Ee p = & AN — pH ee yy, N60’ x r as aon 3 €3 — € + ae eee Sop sin* @ — 2p. cpp nv, ONo yi A+a° € p | Xv AN = pp + N66’. = sin® $ cos? h ( ; sin? d — 2pro*8po)| NO’ Ato . NOp' ioe —(%- ay a psin’ ) (v —" See ee sin’ $) p AN — pH p AN = pe IN GoGo é , 70’ Bt a ; Gi a sin* d cos* \o 6 (« Ses F eS ae sin? $) + AG Vy — = a) = (i). We shall return to this equation later. Thus whether the vibration be parallel to Ox or Oz we get an equation of the second degree in 6p,. Corresponding to each value of 8p, we get an absorption band. Hence we conclude that if one system of electrons undergoes orientation, the corre- sponding bands may become doubled, whether the vibration be perpendicular to the Optic Axis and the Field, or parallel to the Field. This conclusion removes many of the discrepancies which existed between our former results and Becquerel’s observations. In his table, doubling occurs in bands Nos. 6, 16 when the vibration is parallel to the field, and in bands Nos. 4, 10, 22 when the vibration is perpendicular to the field. From the equations for 6p, above it is clear that the expressions giving the index and absorption when orientation occurs are not of the same type as the quantities determining the displacements in the ordinary theory of the Zéeman Effect. These latter displacements are determined from the equation 2g = + u. This may account for the fact that some of the displacements observed by Becquerel were greater than those found im previous observations on the Zéeman Hffect. These large displacements may be due to the corresponding electrons being somewhat loosely held in their orbits. WViOl. 2XCXeNos exehiile 42 308 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS For if the Optic Axis is parallel to the field the separation of the components is given by g=p. OH ey M arore’ Hence if @ is large the separation is large. But a large value of 6 means a small central force, as is seen from the equations of motion on p. 294. Thus if the electron is easily displaced from its position of equilibrium we should expect a large separation of the components of the doublet resulting from the action of a magnetic field. Again, it will be noticed that in some cases a band is not doubled, but is merely displaced or enlarged. If the ‘friction,’ determined by the quantity r in the equations of motion, is small, the maximum absorption deduced from the equation A cide ee ere «. ois xr a 2g +o on p. 800, will be large, and it may easily be shewn that « decreases very rapidly from its maximum value as g increases. Hence, if the friction is small, the bands will be very dark and narrow, while if the friction is large, the bands will be wide and not very intense,—ie. they shade off very gradually from the point of maximum intensity. Thus if the friction is large, the band has much the same intensity throughout a large portion of its breadth, and so, when the crystal is acted on by a magnetic field the components of the resulting doublet will run into one another, and the separation will not be observed. The band, however, on the whole, will increase in breadth. This explains how, in some cases, the bands are observed merely to broaden, without being doubled. The fact that some of the bands observed broaden for some directions of the vibrations and separate for others indicates that the friction is different in different directions, If the friction were the same in all directions we should either get doubling for both directions of the vibration or enlargement without separation for both directions. We have now to explain the fact that in the case of bands Nos. 6, 16, 22 one of the components becomes fainter as the field increases and finally is almost invisible. The systems giving these bands have to be supposed orientated by the field in order to explain the doubling. To find the intensities of the bands we must find the absorptions corresponding to the values p,+6p, of p, where dp, is given by the equations on pp. 306 and 307. Since the values of 8p, in each of these equations are quite distinct, the increments of absorption corresponding to these two bands will be different. OF A CRYSTAL IN A MAGNETIC FIELD. 309 Also these increments will be quite as important as the absorptions deduced from the equation for p,, since these latter mainly depend on the one system with which we are dealing. Hence the two components will be of different intensities. To get a fuller explanation we will return to the equation for dp, on p. 307. We must first state one or two results. We have Nate! = (ty GQ Ab Sa) = tee) =-045n Similarly A+ = a ING For the ‘particular’ electrons X and » are of the same order, and 2’ is large compared with 2. Hence 22’ is large compared with py’, and we may therefore write XX’ instead of Ar’ = pp’. , , s\ Ag AN - { Prcg haraednat] “ P=AN — pp + rosin? +E ig sin? d + oo’ sin? =)X' +o sin? $6 + No’ sin? d+ co’ sin? d =X cos? f + (A +0’) (X' + co) sin? - @: Gee =AXr cos’ p+ a.» gh sin’ d = 1- Therefore if we write K for 2p,c*Sp,, the equation for 6p, becomes Ve («. a a 2) _K (« =. os (« re 21,V> s N@ce atm? ri) m N60’ sin? & be ) € Ar Ar’ Xs - = sin? $ cos? (» = as z : | ( + eed é = " ( fe seas: *) (“2 7 i co) oe) (« aes ses ss gee 7 ee = a sin? cos’ & \@ ( = ee ® ; 5) + (@», - ue )} =0. Consider the second bracket in the second line. 310 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS We may neglect terms in wy’ compared with the last term but one. The last term will be retained since p, p’ are of the same order as X. E (2 Non’ iE _ 5 Ne ¥ 2) 5 (Aa He’ Again &—a=2(> any Te. AX See All the terms in this summation except the last are of the second order in p, p’. F NO pp. ‘ : The last term is approximately equal to — 7 a since AA’ — pp’ is approximately equal to XX. Thus we may replace the second bracket by NO pp’ Bi NO sin* $ fe NO@o'sin?h pp’ ee oe rn’ A NR Now wu, p’ are of the same order as 2 so that = is of the same order as unity. is large compared with Also X is small, so that Aloe Ar’ wy Therefore we may neglect the first term and replace the second bracket by N@c sin? i N@o'sint'd pp’ An Ar’ "NR N@o sin? Np! : ‘ a v, is equal to = “= . Hence since 2X is large compared with X’ we may neglect », in comparison with the other term. aah : ; N6o'sin? hd p’ Similarly in the expression v, +> (pens) A Again in the term », — we may neglect 1. Thus the coefficient of — K may be written (< _ N@o sin’ ) (7 sin? pb ie N@o' sin? S| nr’ Ar An "Ne N°06'co’ . , N*6oo0' sintd pp’ SNE sin? d cos? ¢ + a2 ae Now Tees may be neglected in comparison with «. Hence the coefficient of — K becomes (N@o sin® } he N6o' sin? } ae) o (ay Ni eee Next consider the absolute term. The first line of it is approximately equal to N@o'sin? 6 pp N@c sin’ d €. Dr’ ee NE fe OF A CRYSTAL IN A MAGNETIC FIELD. 311 The second line may be replaced by N?00'co’ . , ao, N@o'sin?d pp’ =r ts Mage aa San © "a i The next line reduces to N@o sin? d Bh N@o'sin?d p’ NOosin? Ar’ 2. AN ee AWN’ : The last line is approximately N20.) ae re est ky a? ¢ cos ba glo £4 ooh, Nea: ery /T N@c sin? $\ i exe Tint hooet BES (a SS"), N°*@cc . 2 ryt or @- a2 - Sin ¢ cos $- 53° N@c sin? ¢ Since « is large compared with , the second and third lines may be An’ neglected, and the absolute term becomes py Nec a . ——— sin, = ? 7dr? : Hence the equation for K becomes N@o sin? 3 N6@o' sin? oy ye pp N*@ao’ . ‘2 2 ‘2 7 7 2° ea ae $s ~ = 0, qi Ke, ( aot yr f) +e, 2 A? SIS which may be written K?— K (2+) sin? ¢ + a8 sin? d = 0, _ V6c sin* p where + - PS N@oq' sin? d BE AN “oe If n,, x, denote the index of refraction and the coefficient of absorption, we have n? (1 — 1K, P = (py + 8? =pre+K 2 = hE K. V4Vo Neglecting the term , we have N ny Q-mya14+ 3424 K 2 a _@ + 8)sin?éd + anes V(a + BP sin? d — 4a8 =1+ >—_— Ne Ne sett) ara smo FE) -Y sin g — doo’ AE = eel AELY (eke) Nl? #9 1+2 x + py 312 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS The imaginary part, determining X, is derived from the terms EN»... | (ee ea Ne, NOsing (oto. Hsing a/(o +o". =| sin? d — 4c0'. xe r AX 2 From p. 309 we may put »’=—g, since 2 is small. 1 es an 1 Also =Sheeste 3, 7 and 5H = Be Therefore the above expression becomes aa yang ee No_ Ne a @ “) sin d + J (1 =) sin? d + 4 x r ee ee 2 : Also A= 29+ th when h= ee 7; 1 We shall retain only the first power of h. Thus we get Ne _—Nésing = (49°— w+ 4ugh)sing t V(49?—2Psin?'d + 162g? + Sigh {(49°—u2)sin?d + 2p} Qgtuh 8g + 12gh” 2 : We shall suppose g to be large compared with A so that terms like 1 ogi may be expanded in powers of h. We shall further suppose (49?— p*)? sin? ¢ + 16y%g? not to become very small. Under these suppositions it is easy to shew that the imaginary part of the above expression is equal to N@h (y—m) sin? d + 2m ee {1 +sing sin ot ear Saray ee 35 {(y—m) sin $ + /(y—m) sin?’ g + rm |} t where y= 497, m= p*. This expression determines the two values of x. We shall now see whether one of these coefficients of absorption can vanish for a particular value of ¢. If so, @ is determined by the equation (y — mv) sin? d + 2m 3 La eS (GS se! =o, V(y— my singh +4ym 27 (y ) —— 1 : 3 : ang t 25, (7—m)sin g + | 1 : 9 : 3 3 E —_ 2 (vy — m)? sin? 9 2 (ean) ay 2 or =e $ +sin? d+ def (y— m)sin® d + 2 7 (y — m) - (y — m) sin? ¢ _ {(y—m) sin* 6 + 2m}? ® Sie (y— mp sin? h + 4ym © 47 | — my sin? d + 4ym} — ay — m) sin? 6 + 2m}, (y y' ry, OF A CRYSTAL IN A MAGNETIC FIELD. 313 1 4m? cos? h ——-l= : : a sin? d (y— m)* sin? d + 4ym ; lies 4m? ‘sintd (y—m) sin? d + 4ym’* *, sin? hd sal ~ (m+ 9) Bm =)" The value of @ deduced from this equation will be real provided that y< m, Le. 29

——. &— yy aa If 7 lies close to a ‘natural’ period 7, of a system of ions we put t= (1+ g)=n +9), and for this system we have X= 29+. = , while for the other systems \ = 1 — r Therefore if &’ refer to these other systems, we have N@ N@ i ecae a ae Se 2 =e 1 a 2g+1.— Bb n?(1—«'P=e+y=14+>’ n?(1— ik’ P=e—y=1 9 Se Mee Do akeeet b, 7] 2 le Hence the absorption for the left-handed wave is a maximum when 2g= 4, and for the right-handed wave when 2g =-— yp. Thus any absorption band in the spectrum is split up into two components by the action of the magnetic field. These bands are circularly polarised in opposite senses. Thus far the development of the theory of the variation of the bands is identical with that for isotropic bodies, the crystalline quality of the medium not having entered into the equations. We shall now suppose the Optic Axis to be slightly melined to the direction of the field and shall find the states of polarisation of the bands composing the doublets. Suppose the Optic Axis to make an angle ¢@ with the field. We obtain the equations of motion of an ion from those on p. 304 by writing 57? for ¢. OF A CRYSTAL IN A MAGNETIC FIELD. 315 Thus the equations of motion are. wile = 906 , eH On eS Tc Sr ers govt ane ,0n eH o& 4ire? Are? ™ > eY — (5 sin’ 6 + “2 cost) n — re BT oat 788 #008 GE (Fe — 7) > 4ne 47. 47e 47 m oP = ef — ( 7 008 = sin’ 6) € — sin ¢ cos by (> - =) ews t If everything be supposed to vary as e”, we derive from these equations exactly as on p. 304, the equations 0 e&X — ven = a aXe: en (X+ 0° sin? g) + eo’ sin g cos Hf + vewé = = Ve T ef (X’ — o sin? d) — eosin ¢ cos bn = = Z, where the quantities involved have the same meanings as on p. 304. Correct to ¢? these may be written 6 e&X — cen = — X, 47, en (X + a $*) + eo GE 4+ vepE = = We ef (' — of") — eadn = — Solving these equations for &, », ¢, and neglecting powers of @ above the second, we get for the components of current CG OMe r) dary 2 = &y Ot + uv, OE +t.op Te ,0x Os dary =— wy a +e : op Tx: oF ox oY VA dares = — top = +x. OE + & — aE? AN + (No — Ao + aa’) ¢ where eee NOP p = 14209. ~— 28, e=14sNe iia ae p n= SNe. ~— oe pan Me. es 2 6 =— 62". p p T Ppa Coeete pas Sais eee p p p= 02— pw) + (ANG — od? 4+ op? + doo’) G*. Vou. XX. No. XIII. 43 316 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS Consider waves travelling along the axis of z. Then the quantities involved are functions of z, ¢ only. Putting (X, Y, Z)=(A, B, C) e "?? and substituting in Maxwell’s Equations as before, we get (A+u/B+tapC= A. pc? —~w{/A+e’B+y.¢.0=B. pe —woA+yB +6,0=0 Eliminating p and C’ between these equations, we find fea Se 9) + OSes eo vy — wx? From this equation we derive two values for the ratio of A to B; complex in general. If either of these values be substituted in the immediately preceding equations we get a value of p® to correspond, giving us the index and absorption for each component. The states of polarisation of the components are determined by the values of the ratio A to B. Thus when the Optic Axis is parallel to the field, we have ¢=0 and A*+B?=0, or A=+vB, indicating that the components are circularly polarised. € (4 — €) +(x? = 0°) v6 — oxd" : We have Bae = INAS eee p Now ¢ is small, and for all systems of ions except that under immediate considera- tion A, ’ ete. are not small. We must evaluate Hence for these systems we may put p= (= p’) =A? approximately ; m being small compared with 2» for these systems. Thus if >’ refer to all the other systems, we have Na: No’ +0’ Né.——“— @. arn = ¢ eg —« =¢? >’ NGc’ Suppose that the impressed period is almost equal to a natural period of the system we are considering, and put t=7,(1+g)=¥Vb,(1+g) where g is small. Then for the particular group of ions =2 @ Xr ee ts From p. 309 we have rv oe 0 OF A CRYSTAL IN A MAGNETIC FIELD. 317 For the particular system of electrons considered X is small, and we may sometimes use \’=—o as an approximation. For this group of electrons p = (A? — p*) (X' — a?) +o’ (N' + o) 6 2f/ _’ =-o (i) (14 42) +82 ¢° approximately = —a(*— uw) — Nog? approximately =—o(*—.p*) approximately ; re’ = Neto Gh —¢"=¢°>'N0 ——, — N@c’ . ————. ¢? part ee eOron)? Xx tO r = v2 'Nbe! *P2 ye. > = ¢°>'NG@0’ says AY) vie? since Oo’ =— Oa. Again iS Nee eee p =e — > (5 x Eee) 2 V’ - ; NO her lesa where €& = yo? ; M+ pw - = = >N@ ——__——— € o x’ o) =a-¢ S N Be Xo teins, N@ ne : $° r IN ft} ny N=—B NO0c o—xr X =e — ¢. >’ NM ase rn? r oP sd Xr ust Mig aol = SNe CoS p (5 | rN — od? =»,-— Nye £ ) ee HEN a p =y,- > a Rene) P* ld p * 6 ow Nur’ = +60 NpOX0o’ AY the. pa ae re ae Tara a a Ol eee Oe) NpO@r0c dN4+oa a Ne =y,—> ; —~, 6. — .@ ee, OF et eee Np Np Npéo’ No IN = Gee a ae its nr = x2 Le are = XC Tae ¢ Nu8 . cae ap? 318 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS > Nyc Np p N(R) _y Npéc Npé = a 2 2B 2 _s Nr€ca - p wv N@c0 =a NXE * Taos = yer 5 . Ve “2X NOX 1 @ — wyd?= (a-¢. > ae a = = $) ,Np@ Np os NuO0' ee eee eae De B 2 ee V . 3 — Nuéd 02-2) ~ Npbc Nu (s’ N@c Nr : -(2. 565 —— (3. Sor = ) ge The terms in the expressions for 7, €', etc, which depend on the particular system of electrons, become complex when the impressed period approaches the free period. We may suppose the real part of these terms negligible in comparison with the remainder of the expression for »,’, ete. Thus we may regard the quantities »,’, etc, as made up of a real part, and an imaginary part which is small compared with the real part. Hence we may write haat)

Ant (63 Ne: : MG as nO 5 NGc o x nv r»AS Te ee ee ee oN Nyéc ., NOc ee ee) 2 Sal qa x)? = PAG G, where G is small compared with ¢,> If it be remembered that ” b t= Gre ae ik me) b ee ye tae Ty b b’ =1-;, =1-;, so that oc, o’, X, X’ are all fractions of much the same order of magnitude, it will be seen that each of the three terms : : : , : , Ne composing the coefficient of ¢* in (1'e.’— wyd¢?) is of the same character as ¢,= — : OF A CRYSTAL IN A MAGNETIC FIELD. 319 Hence we may suppose this coefficient of ¢° to be fairly large compared with ass ale a2 a Thus we may write v, 6 — oyd’? = E — Fg? + 1G, where F is large compared with # and G is small compared with H#, Again, owing to the presence of the factor p, o is small compared with y, and will be neglected. Neglecting the small imaginary part arising from the particular system considered, we therefore have Py 4h i F bie a NO’ N+c0 (a, NOc\? ef (6! ~ 6") +00!) = 9 ad’ Se. SE" + (3.37) ae UN = D¢’, say. Therefore we have é (q —& ) +(x? — @”) d? _ ly $? Vy. — wy? E-F¢?4+.G"’ where D is large compared with #, since # contains the small factor p. When ¢=0 the state of polarisation of the bands is given by A?+ B?=0, Thus the bands are circularly polarised in opposite senses, As @ increases, the coefficient of AB also increases, remaining almost purely imaginary, since G is small com- pared with Z£. But now suppose that @ takes one of the values given by #— F¢?=0, Then the coefficient of AB becomes equal to 76 which is large and real, Ih : Al Thus the equation giving Rs At 7g? AB+ B=0. : Dee : A Since G ¢° is large and real, it follows that the two values of BR are real, Hence for this value of @ the two components are plane polarised. When @¢ changes from the value + 2 this state of polarisation abruptly alters from plane to elliptic; for the term vG may now be again neglected compared with H — F¢?, and thus the coefficient of AB again becomes imaginary. eA. : The ratio = is now given by B A? — 1k. AB + B?=0) k real, F A_tktV-k?—4 Sie am kt Vhe + 4 = eS 2 320 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS This shews that the two bands are now elliptically polarised in opposite senses, since k+Vi*°44 is positive and k—Vi?+4 is negative. The value of @ for which we get plane polarisation is given by ea%, Wika where f= 2 pace Bile Nite Mal yMOe CoN, y Nate Noe x rn 7 2 Ni VX Xr Since #£, F both contain throughout the factor ~, which depends on the field, it follows that @ is independent of the strength of the field. Moreover, ¢@ will be practically the same for all the bands, since the summations which occur in Z#, F refer to all the systems except the one giving the band. According to Becquerel the direction of polarisation of the bands is independent of the strength of the field. We will now deduce this from our theory. We shall also shew that the directions of polarisation of the components are perpendicular. The direction of vibration depends upon the term G. The imaginary part of »,'e — wy¢? is contained in Nué = ae o —Xr Won ee . 2 roar ea eo) tae OE ae aN HO6 Gam ap Wh $4 ane es Bey / —— — es Se = = NN? a $ (A° — p*) O28 = 2 The terms + ae ae pe cancel. GQ Now A= 29g + th where hail, and the positions of the bands are given approxi- 1 mately by 29=+ y. (Their positions will be altered by an amount depending on ¢%*) Therefore X is of the same order as p for our particular electrons. Hence it appears N, : = that the term = ues € is the only one we need retain, for the other terms are all small Yala compared with this. a Nye _Nypeé Bey aw (Qg+ a = 8 If we take 2g=y this becomes oe or €, and if we take 29=—y it becomes 5 “g=—p : 2iph OF ve +e oh €5. Therefore we have G= eS and G is independent of the strength of the field. [It is to be noticed that whether the above is the only term to be retained or not, G@ will be independent of the field strength, since this will cancel out in each term.] Thus the direction of the polarisation, which depends on Ae i is independent of the field strength. OF A CRYSTAL IN A MAGNETIC FIELD. 321 : B, é Again, suppose that a: A, are the values of a for the two components. The 1 42 positions of the components are determined very approximately by 2g=m and 2g=—yp. For these values of g, G takes equal and opposite values. Since the two values of @ are of opposite sign it follows that a and a. are of opposite sign. 1 2 Further, since the values of @ are equal in magnitude, it follows that if we reverse : : ; B, : : the sign of one of the ratios =, ap they will both be roots of the equation 41) + 2 N6 aS «. AB+ B= / B, BBs Hence we have (2) (- = = il oF ri 2 z == 1. If the normals to the planes of polarisation of the components make angles 6,, @, with the axis of « we have B, 1345 aaa Ga asap (rhe *. tan @,.tan 6, =—1, 7 aie Gnew ta Hence the bands are polarised in perpendicular directions. Again, the direction of polarisation will turn through 90° when the field is reversed. For the displacements of the bands are given by 2g=+ xu. Thus the bands change places when the field is reversed, since u changes sign ; the component that was accelerated being now retarded, and vice versa. Hence, since the direction of polarisation of a given band does not change with the field, it follows that the direction of polarisation of the band which occupies a given position turns through 90°, since the bands are polarised in perpendicular directions. The results thus deduced for the states of polarisation of the components when the axis is slightly inclined to the field agree fully with Becquerel’s observations. He, however, observed that the angle ¢ depended on the thickness of the crystal. The above theory seems to leave no room for such a dependence. With the Optic Axis parallel to the Rays and to the Field it was found that in a large number of the resulting doublets the direction of displacement was contrary to that observed in the Zéeman Effect. According to our theory the displacement for the right-handed components is given by 2g =— yp, and for the left-handed by 2g =+ yp. Thus for these ‘inverse’ displacements the sign of ~ must be different from the sign for the direct displacements. Now a a Aqrite- 322 Mr PAGE, ON THE VARIATION OF THE ABSORPTION BANDS, Etec. Hence w can only change sign when either H or e changes sign. In observations on the Zéeman Effect it has always been found that the vibrations are due to negative electrons, and vibrations due to positive electrons have never been observed. From the separation of the components of the band whose wave length is 6424 yy Becquerel infers that the vibrations are due to negative electrons for which <= 115A But for the band whose wave length is 52214 he finds almost the same value for <, but considers it due to positive electrons. oe e The coincidence of these values for = leads us to suspect that the electrons are, however, negative in both cases, for it is well known that the ratio < is approximately constant for negative electrons. e ae Also values of = for positive electrons, so far as known, are always much smaller than this. Hence it seems preferable to conclude that the vibrations are due to negative elec- trons in all cases, and to suppose that in the neighbourhood of the systems which give ‘inverse’ displacements the magnetic field changes sign, i.e. that these inverse effects are diamagnetic in character. M.: Becquerel* and Herr Voigt+ have recently given explanations of the observed phenomena, starting from a hypothesis rather different from that used above. * J. Becquerel, C. R., 19 Nov., 3, 10, 24 Dec, 1906. + W. Voigt, Gatt. Nach. 1906, Heft 5. XIV. On the Asymptotic Approximation to Functions defined by Highly-convergent Product-forms. By J. E. Lirrnewoop, B,A., Trinity College, Cambridge. Communicated by E. W. Barnes, Sc.D. [ Received, June 3, 1907.] § 1. In the present paper we shall develop certain methods by means of which it is possible to obtain asymptotic approximations for certain classes of functions defined by a product-form of the type II j1+( s=1 ;) al In this form we suppose that a, d2,... ds, +. is a sequence of numbers arranged in order of increasing moduli, and that |a;| tends either to an infinite limit or to a definite finite limit when s tends to infinity. In the former case |z| is supposed large, in the latter |z) is supposed less than, but near, the limit to which a, tends. The earlier part of the paper is devoted to certain results which hold only in the former case, while in the latter part the two cases are treated simultaneously. § 2. Throughout the paper we shall employ the following ideas and notation. 2 F (s) We write F(z)=l [2 +(=) i @,\=@,, |Z2\—=n7 s=1 8/ The index f(s) is supposed to be real. When f(s) is an integer for all values of s, the function F(z) is uniform. When f(s) has non-integral values, F(z) is multiform. In this case we suppose a cross-cut made from the origin along the negative real axis, extending to infinity, if Lta,=2, in which case z ranges over the whole plane, and s=n extending to the point—c if Lta,=c, in which case z is supposed interior to the circle 8=n |z|=c. Then in the cut plane 2/® is interpreted to mean exp[f(s)logz], when logz has its principal value, and the function f(z) is uniform in the cut plane. When Lta,=x, we propose to find an approximate expression for F(z) when 7 is s=n large, and when Lt a,=a finite c, to find a similar expression when r is less than but s=a nearly equal to c. In either case there is a single integer associated with 7, which we shall always denote by n, such that a,.,>r>4. Vion; XexXe) Noy Xalve 44 324 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO When r is near its limit, » is clearly large. The approximations which we shall obtain are expressed practically in terms of n, and it is the fact that n is large which enables us to deal with the case when Lt a,=c. s=n We shall always use the following notation [2 n P= S70)| . log z— | 27@) log a,| ; s=1 s=1 ea aA f(s) x j= = log [1 +(2) |-3)0 99 aaa — + (ate =| ae ate coy bd = (Li s=n+2 Us 5=0) n—1 by F ts) n-1 Gh SF (n—s) 8 ="S tog [a + (%) ] =73' tog 1+(“) : s=1 \ 2, s=1 zy f g \f (n+l) 1, = log |1 + = i : Unti ; Ff (n) T,=log E i (*) | We then have, as is easily seen, log FF (2) = P+ EARS aaa ae Dig c. te ses cnacnccnseesceeoen eh eeteae (1); We shall also denote the real parts of P, 7,, ete., by P, T., ete. It is customary, and it is more convenient, to express such results as we propose to obtain, in terms of the function log F(z) rather than F(z), but we are none the less primarily interested in F(z), Now log F(z) has an infinity of values, differing by multiples of 2m, and when log F(z) is given, each of its values gives one and the same value of F(z). For this reason we shall regard all possible values of log F(z) as equivalent, and we shall therefore not concern ourselves with the proper multiple of 2 which should be added to the right-hand side of equation (1). Again, the imaginary part of any approximation which we may have obtained for log F(z) may have a modulus which tends to infinity as r tends to its limit. But since this imaginary part may be reduced, by the subtraction of a suitable multiple of 271, to an expression whose modulus is less than 7, and since it corresponds to a factor of F(z) whose modulus is unity, we shall not regard this imaginary part as being a large term in the approximation for log F(z), but as being a finite term. After what has been said it is clear that we are free to choose any determination we please for the logarithms which occur in the above expressions for P, R, ete., provided only that the infinite series for R converge. We determine that all the logarithms in the expressions for R, S, 7; and 7, shall have their principal values. The logarithms in the expression for P we shall choose as is most convenient in each particular case *. This is clearly legitimate on account of the nature of equation (1). We shall, however, sometimes employ the two following variations of equation (1) log F (2) = P+ RIBS 4 TE De tye cemeenctveesas.cs0 ae (2), and log F(z)= P+ R4+84+7%4+7.4 vy, where we may suppose that |y.|<7, yy, <7. * For example if a,=se*Y', it is clearly most convenient for purposes of calculation to take loga,=logs+sy.. FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 325 § 3. We shall first establish a general formula for log F'(z)=log II [+2], as under certain general conditions for f(s) and the a’s. We shall then consider more special forms of f(s), and shall replace the general conditions by others more precise. Theorem. Let a, a... be a set of numbers arranged in order of increasing* moduli @, %.... Let f(s) be a number depending on s, such that f(s+1)>f(s) for all values of s, and let the following conditions be satisfied, (A) Lt fee = it] Lana (B) Lt ie (log 2,1, — log a) = 00. (C) Suppose we are given any finite integer p. Suppose 7 is a positive number, \F (8) and let n be the integer such that a,<¢r log|1+ (*) | 5 El GAs, Ns [1 or () : = Z =m’ Zz s=qtl s=m'+1 where m’ is the integer provided by condition (C)(2). From the condition (C) (2) (a), we have, when m’>s>4q, ds 7 (8) () Fig) = B.(r) <8. Bra (n) <0. (*#) aakee ee (7) a\ F(a) a, | log | 1+ (2) | | Zz | qt+1 F (q+) m! <2. (2) 5 Ga Lt s=qrtl 2 f f < ie (941) (CU yd (Gt oe eit dale eic'els wis isisia cement (9), where, from the definition of g, we have f(q¢+1)>>p. FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 329 We have finally to consider o,. Tt is evident from the inequalities by which m, m’, are determined in (C), that m>m’. Now: by (€)(1);” 6.7) >/8s@)> .-: > Bar); Bm (7) < Bm) <<... < AO: Then when n>s>m’, | /a Tis) ’ yn’ Jim) a, T(n) | (“) = 8,(r)< the greater of (*) and ea Ldearsataneeeds (10). é /a,\ fm /Q,\ Fim’) First suppose (=) > (*) . a,\ fin a Jin) ia Ef (n) We have (=) <( ae Seas ) r Anis & An +1/ paw) =k 1 La ee) fh ster eA BUR or oLE (11) < angi °°”) <4.when n is sufficiently large ............... (12). Hence, when 7 is sufficiently large, n \ Fla) lol|< & |log E (“) | s=m'+1 [ Tin) = <(n—m —1).| log a = (*) | , since ie a ee! — A —- =|2 Se = = fin) <2(n—m'—1). = , from (12), x(n) ii ae from (11). EE ere 4 -k < 2n.dniy *™) /(1+ Therefore ke log [r-?/| a, ]> ze log n+l xo > Lt | [A = 198 Ori -~ log f n=a 1 ee re since aA tends to infinity and is therefore ultimately greater than 2p, x (n) + 7(n) 24k Lt | f(x) logan../1— sore lop ae “ n=0 ee 1 x(n) k F(n) , | f(r) (log O41 — log ay) ee log 2 2k ee | m8 2 { log n k klogn =+ 0, by condition (B). Hence itor |e —=ce((t) wheres lithe) SOWcrcnensesenceresseonessnsecse (13), Gok Jim) ‘a Jin) Next suppose = = (=) : ; = 330 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO By condition (C)(2) (b), ee OO) Tie pe TE en (Am\ Fm 1 Choose n so large that (72) <5: Am! idl Then as before o.| < 2n (“= Am! Fim’) Fi and therefore Lt [7r?|o.|]<2 Lt E = .r?| =0 by (C), (2), (d). Hence, in any Case, Gs oT! —e (Qr), elite (72) — Oe eamoen ences cesesenasenetescaeen (13yY. n=2 Then remembering that S.=o,+¢., we have, from (4)’, (6)’, (9) and (13y, wT z 2 r? | 7.4+R+7,4+S— > C,/22|<407+ K.P? + Tee (gai: ) 79D PFD) + €(n) s=1 pa =e(r), where Lt e(r)=0, since (« — p), [f(¢+1)— p], are positive and do not depend on n. Since p>v,, we have Lt tog F(z)-P-= oye" | Set |S 1b [tog F(z)-—P-— = ce") 2?) =0, TH=D s=1 r=a s=1 and the theorem is proved. § 4. The formula of the last article supposes z confined to the region R. It is easy, however, to see that when z is unrestricted, the expressions R, and S, satisfy the inequalities that we found in §3 for 7,+ R, and 7,+58. Hence when z is unrestricted we have the asymptotic formula, log F()=P+7,+T,+ = Gl, s=1 n+ riod Moreover at least one of the two expressions 7,, 7., is asymptotically represented by zero. For if z lies in the annulus A,, 7, and 7,* are both asymptotically represented by zero. If z lies outside the annulus A,, in such a manner that A, passes between 2 and the origin, | Z,| is less than the corresponding value when z is inside A, and has the same argument, and hence 7, is asymptotically represented by zero. Similarly when z lies on the other side of the annulus 4,, 7, is asymptotically represented by zero. The above extended formula enables us to study the behaviour of F(z) when 2z is near the circle 2 =/a,. an Sin) * In the course of the proof in § 3 we proved that Lt n.(@) . 7? |=0, whence follows the above result for T,. n=x FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 331 We may remark, in connection with the conditions (A), (B), (C), that (C), in spite of its complicated appearance, is usually satisfied, in the cases which arise most naturally, when (A) and (B) are satisfied. Moreover in the case of any particular function, it is usually fairly easy to determine whether or no (C) is satisfied. It is clear that when no further assumptions are made as to f(s) than that /(s+1)>/(s), we must expect complicated conditions. We shall now assume special forms for f(s), and shall replace in each case the conditions (A), (B), (C), by other and simpler conditions. § 5. Suppose that f(s)=s. The function F(z) is the integral function Paheacane We proceed to show that the general conditions of § 3 are satisfied, if the two following conditions are satisfied : log asi1 im E = x (s) = = 0 | ee -1] =<. Il. @ (s) = log as, + s (log 454; — log as) is an increasing function of s after some finite value of s. Consequently, under conditions I, 1], when z is confined to the region R, we have the asymptotic expansion, ee C,/2, where P=4n(n+1)logz— = slog as, 1 s and =C,/z° is the formal development of log II E + (“) I; arranged in descending powers 1 1 Zz of 2. The condition (A) is identical with I; we have then to consider (B) and (C). We can deduce from I that when s is sufficiently large, a; > exp (s‘), for any assigned value of 2. OF Gs14 Let » be the least integer such that xy(s)=s Hes - 1| >2A+1, and also a,>1, for s all values of s>y. Then log te =| 1+ ¥O) toga, =... = log a. ( 14%) (1 )Q+ + XEeh). (14 %2) A+1 +1 vgs (04222) (14824) (02 ae ii 1 (/ r+] > Toga, exp | 41) (5 +525 + +7)]- |i |@+™4) exp -***)|} / = | : Now the product in crooked brackets, when II is replaced by Tl , is convergent. Hence the modulus of the product in crooked ache: lies between finite non-zero limits, and has a lower limit K, say. Vout. XX. No. XIV. 45 332 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO ; 8.1 st dla: s+l1 Se os — Again ae ie ; ] ea A+1 Then from (1), log a4, > K .loga,.e w.(s + 1)". A+1 Choose s so large that (s+1).K log a.e ati > 1. Then log a4, >(s +1). We therefore have — los: : en rs [Aomaen=oee) 5 ry [lee yo] > me [pox] =e. ise log s Gacy (LOTS since Lt y(s)=o. Hence condition (B) is satisfied. We have finally, then, to consider (C). We have B;(r)= exp [s (log a, — log r)]. Ee Bsa (@)_ SPs Then ys (rT) = Bey exp [(log a4, — log 7) + s (log 4,4, — log as)] =exp[(s)—log 7]. Now by condition II, w(s) is an increasing function of s after some finite value o of s, When r and m are large w(s)—logr is negative when s 0 when s>m; that is, such that Bers (7) 1, when s>m; or again, such that 8,(r)>A.(r)>...>Bn(r), and Bm (7) < [stan (r) < [shes (7) <...< By (7). Thus (C) (1) is satisfied. when sm’>p+41, when n, or 1, is sufticiently large. Also we have n e”. Am! i On44 Lae r Hence Lt |n. (*' o7P)| < Dita ae (“ee) KAR a r —s r “ loglogr > n, We determine a number a by the relations, 0 C,/2, where C= e, C= ee gy der", C= ee = he? C.= ee = pene + te% ete. When z= a we have the simple expression 2 1 1 log F(z) = Je * loge zy. log z— E + =| log, 2. log z - Sey g + log 24% C,/2 (er— 1? (cok) ita § 7. Next consider the case f(n)=n!%, q>1. The theorem for this case is the following. The general conditions (A), (B), (C) are satisfied provided the following conditions are fulfilled :— x log as 44 ae (a) a x(s) = Be sf [eae - 1] =; st (8) ae log s [log @541 — log a,] = 20 (y) wW(s)=[(s +1)? log a,.4 — s? log @,]/[(s + 1)4— s%] is an increasing function of s when s is greater than a finite o. (6) One or both of the following conditions are satisfied :— (a) A positive non-zero constant p exists such that when s is sufficiently large, a, > 5°. 45—2 334 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO (b) A positive non-zero constant k,<1, and a positive non-zero h, exist, such that if s be sufficiently large, and if s’ be any integer less than h,s, then [(s’ + 1)¢—s'7] log a, —(s' +.1)?. log ay41 +8 log ay >h, The conditions (A) and (B) are here identical with (a) and (8). It remains to consider (C). We have > / (s+1)? st Been) — (Beet) 8 [(B)'= exp lee + Dt sth 4(6) — 10g 1) eee (1). The expression in square brackets is negative when s. Then it follows, as in the last article, that there is an integer m such that [SHY SVEH@)S con S/S AO» and Bal?) ay/(p + a , where p is a number provided by 6 (a). When 7 is large we have m>m’. Bsir(r) From (1) we see that when r is sufficiently large, then, when s n. y= 7 7t== 00 ppt n=a2 An Then (C) (2) is satisfied. Next suppose 6 (b) holds. Choose m' to be the greatest integer contained in hyn. We have from equation (1), a) < exp [ — {log a, [(s + 1)? — s2] — (s+ 1)? log as41 +s? log a,}]. Then from the condition 6 (b), we have, when s< m’, Bs+1(7) —h B.(r) ~< ert < ji A ; ky Again m =[kn)>hn-1>sn, when n> ke 1 greater than unity, but not otherwise. F(n) =[n1], g>1, where [a] is the greatest integer contained in a. (6) be modified by replacing s?, wherever it occurs, by [s?] it can be proved, by slightly modifying the analysis of § 7, that the general conditions are satisfied, so that we may still Gy, Ag, «ee FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. Hence (*) < (= ees , SINCE Ay <7 ky)? 4, 2) ni 2 (*\ 2 [rs (5) -m >] .nt 3 - Pp < | ae > , since Zz lies in R a, An+1 i} a. \ka.nt k\@ < = |) , where h=k (3) : An +1/ 2 me? a kant Then L= Ut log [ay n (=) | > Lt log [rm (=) 28) y=oa n=2 An +1 > Lt [An?. (log a,,,—log an) —p log any. — log n] n=w (n) SsLib Ig. “7 — log Or — p log aria — log n n=x 1 7 xX@) nt Now Lt = 2, since Lt y(n)=a n= aes N=2 eee Hence when n is sufficiently large, p <>. axl) x(n) * ue nt Hence eS Vit iy axe log a4, — log n Seo || 7 1 4X) nt > Bo [4h .n@ (log ¢4, — log an) — log n] zu t | ice : Ss ;, (log O41 — log ay) — 1} log n|= +a, since, by condition ({), Lt oe (log @nii — log an) =+ 0. Hence in this case again (C) (2) is satisfied. Thus (A), (B), (C) are satisfied when (a), (8), (y), (8) are satisfied. 3390 § 8. The functions considered in the last article are integral functions if q is an integer apply the general formula. We obtain integral functions, however, if we take If the conditions (a) to With regard to the conditions (a) to (8), we may remark that they are satisfied when In support of this statement we shall make the following observations. is any sequence whatever whose terms increase to infinity in a regular manner. 336 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO In the first place if a, %,... increase in a regular* manner to infinity, we have Bo : 0g di — loga, > Se ; 1 For if not we should have log a,4, — log Os < she? for some value of ec. Then, adding, we have i 5s aultte 1 1 OD ar < ae 1, that conditions (a), (8) are satisfied. It is, moreover, evident that (y) is satisfied when a, %,... Increase in a regular manner. Concerning (8) we observe that with our suppositions, either a, >s° for some p and all values of s sufficiently large, or else a,h > 0. If s’ is supposed less than any particular finite limit, then, by choosing s sufficiently large, we can ensure that the above inequality is satisfied. We shall suppose, then, that s’ is large. row Let O(s')= ds’ (log ay). : 1 Atel ee Pe ee Since log ay = ae log s’ is of less order than logs’, @(s’) is of less order than ©. t Let (8) = sg and put s’=e* Let $(s’)= w(t), and let | $o72 We clearly have Lt W(t)=o. t= SH bo an Then log ay = | O(a’) da’ = uty = ((G): Since Lt logay=2, we have Lt O(t)=o0. t= Then log a,—log ay >log avy —logay, (k’ =1/h,) > O(t + log k’) — Q(t) > log k’O'(t) + E(log KPO" (t+ A log kK’), NL cceeeeeeeeeeeeees (1). Now 0(t)= a is positive and decreasing, and hence {1’(t) is negative, and decreasing : Sake ; OM) _ in absolute magnitude. Also ue OO) =0, * The word “regular” is used above in a rough sense, and the analysis following is only rough, FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 337 We have then from (1) [log a, — log ay] > (1 — e) log k’. O(t)>(1 — €) log k’. a Saasiaaceees (2) oa (Cle Te lo mdiceersiO (Si)emec eenecctesseincon tes. naeceateeeeee: (2)r Again log ay, —log ay =[1 + e(s)] @(s’), since O(s’)= 2 (log ay), and tends to zero with 1/s’. log ay SBC y l+e 1 a = ( ) 57, Whene<5. 2 Bfeaee log a, — log ay > Gt —e)logk’.s’ es log k’.s 3 Then wW(s', s)=[(s' + 1)2— 82] (log a, — log ay) — (s’ + 1) (log ay, — log ay) = /q-1 y - > (log a, — log ay) |" ee etl) log k’. =| 0) > (log a, — log a) | gs’? — 24, 57, - —). (0g 4 2. da log k’ aS QaH1 Choose &, such that SS GE log k ; Then vr(s’, s) > (log a, — log ay). - OLE > (log ayy — log ay). Z aS > (1 — e)logh’ fsa (from (2)). ee #0 Now 20=[F5 tends to infinity with ¢. Hence we must have y(t)< f < (logs’). Th 5 E ; ses en ee) ene EMO eae /q-1 But since g>1, Lt as a x», and hence the inequality which we wished to prove is seen to hold. nee) § 9. As an example, let us consider the case a,=n°, where Rp>0O, and n? means exp (p log n). Let p=pit¢ep.; then a,=n". It is easily seen that (a), (8), (vy) and (6) (a) are satisfied. We may, then, apply our general formula. The region & is the sum of all the annuli (s+ 1)Fr. sD < | z|<(s+1)0-Ver, shor, It may be taken to be the sum of all the annuli om. (14 ‘) <|z|r>n4, or n+1>72>n. We take m= rie —|@, where: Ol s? may be asymptotically expanded* as s=1 1 C = dst as SS (i) ea & Dea” see Del NS at dy where Lt | Jz.n2+|=0. n=D Moreover this expansion may be differentiated with respect to q. n oO % : Hence logn. = toa y st=C(—q). logn+€(—q)+- s=1 nin s=1 1 (q+1) : C; SESS g+i¥r +r ee S,(1|1). x |e i] Kk, where Lt | Kyn-2|=0. Hence from (2), l log F(z) =p Be mY log ray Ee q) at G rESY, nin 4+ > Lope oo] r=1 2 + [log z — p log n] . {t-o+a9 ny Ps S,(1|1). ate nin ‘ ” + > C,/23+ H,+ H,’ fae ATT, HT? a oe (3), s=1 | n2—-q-1 | 3 where Lt =(0, and Lt | H,’. 2v|=0, mes * log n wes and where+ BIS doe a (222) meg gees and =C,/z¥s is the formal development of log. i E ats 2)" ‘ The formula (3) provides an asymptotic expansion for log F(z) in descending powers (possibly non-integral) of z, and in descending integral powers of n, or of [7 — a] * Barnes, Proc. Lond. Math. Soc., Ser. 2, Vol. 11. Part 4, p. 262. + We clearly have By=8; (1). 2 )=0, and Bo,;=0 (s>1), since S2,_; (1|1)= FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 339 The obvious objection to this expansion is that it involves the number a. This, how- ever, is unavoidable. If we expand the various terms in in descending powers of 7/9, and arrange the whole expression in terms in r of descending order, neglecting, to our order of approximation, pure imaginaries, which count as finite terms, we obtain, when q > 2, pO 6a(lL—a)—1 2% = log F(z) = rh toy. r?i +A. + (lower terms) ; (q+1) 12 when* g=2 log F(z) = = PPL +t p,. Sa eee rp + (finite terms) ; and when 1< q< 2, qt+l > gq} Ae ef Ge(ILSey= il ; : log F(z) = Ge 1’ P+ py. i eee €(— q) log r + (finite terms). It is clear that these expressions cannot be expressed in terms of the ordinary analytic functions, since the function @ cannot be so expressed, and since it enters inevitably into the above formulae. When z=n°, so that a=0, we can find an asymptotic expansion for log F(z) in terms of known functions. nt We must add to the general formula P + =C;,/2"s, the term log [2 + (*) iF whose value is log 2. ; We then have qt+1 neg pz? 4 = q+1-2s 0 Z)= ea -_ Bagge te 8 (2) (q+1P ee 23 7 + €(—q)logz+pf(—q)+log2 + = C,/2s+ H,+ Hy’, s=1 2l-q=-1 where Lt |H,.2 ° |=0, and Lt |H;’.2v|=0. pet T=a $10. By means of the function F(z) of the last article we can give an example of the following possibility. It is possible to construct two integral functions ¢,(z) and y,(z), whose moduli differ only by a factor of the form [1+ A/z], where A is finite, on the whole circumference of circles with centre at the origin and radii as large as we please, which functions are such that every zero of ¢,(z) of large modulus is at a large distance from every zero of y,(2), or more precisely, given any number h, as large as we please, we can find a number & such that every zero of ¢,(z), which is outside the circle |z|=h, is at a distance greater than h, from every zero of y,(2). oo 2p \ 8! Let F,,p(2) = 1 E + (3) | * We have ¢(—2)=0, so that the term ¢(—q).logr, which appears in the last expression, is in this case absent. Vou, XX. No. XIV. 46 340 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO where g is an integer > 2, p is an integer >q, and p is an odd integer. F,,,(z) is evidently an integral function of finite order. From the definition of the region R, it is evident that this region is independent of p. Clearly also, for a given z, the integer x is independent of p. For the function F,,,(z) we have nu P=plogz. = s!—pp = stlogs s=1 s=1 =pl’, where 7’ is independent of p. x“ a) pp s# Hence if = ,C,/z* be the formal development* of log II E =P (S) Ik ay ~ s= e=1 log F,.n(2)=p7l+ = ,C;/2, s=1 and log | F,»(2)|=pT+R[ = Cy /2], where T=RL............000008 (1). s=1 We have also the particular case log.| Fiip(2) | = 2+ ay et xees ayn (2). s=1 Again, $(z)=F,,»(z)/Fi,p)(2) 1s easily seen to be an integral function. Let an (2) i= [ie (2) Pees Then we have from (1) and (2), when z hes in R, log | @(z)|— log WZ) = x = Ce SPs 1Cg). 2%. Thus | $(z)| and |y(z)| differ only by a factor of the form E ~ =| where |.A| is finite. Now the zeros of $(z) are n” . exp & (2¢ + 1)| On where F— On Ve — 1) Ss AE ier) (al) en 7) sees restate The zeros of W(z) are n” exp S (2t+1), each zero being taken (p—1) times, where HSS IMisca (Cal, eral et, 2 BYosac When nx‘ is not a multiple of p the two sets of zeros on the circle |z| =n are completely distinct. When x‘ is a multiple of p the two sets have common members. Let the assemblage of these common zeros, for all values of n, be 0,, )..., arranged in order of increasing moduli, each common zero 0,, being taken ~ times when it is a common zero of order » for both $(z) and (2). Let G(z) be any integral function with the zeros 0,, b,..., and no others. Then ¢,(2) = $(z)/G@(z) and W(z)=wW(z)/G@(z) are integral functions, and their moduli only differ by a factor of the form E + al : * Since g and p are integers, this development proceeds in descending integral powers of 2. FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 341 Now on the circle |z|=n”, the least distance d(n) between a zero of ¢,(z) and a zero of ¥,(z) is greater than or equal to the least distance between two distinct points of the set 2sre n? exp [zrae+ 1) | e°, where t=0, 1... (n‘—1), and s=0, 1, 2...(@—1); for every zero of ¢:(z) and w,(z) belongs to this set. Now the difference of the arguments of the two points of this set for which s and ¢ have the values s,, ¢,, and s,, t, is 2(s, — 82) 7 E 2(t zs ty) 7 = ((s1 = 8.)n? + (t, = te) p} Dem P nt prt The expression in crooked brackets does not vanish since the two points are distinct, hence this expression, which must be an integer, is equal to or greater than unity. : 2Qar Hence the difference of the arguments > ce pr 5) . . : aT Sete : Then the distance between the two points er at ona? and becomes infinite with n when p>q. Again the shortest distance between the circles |z =n”, and |z|=(n—1)?, is greater than p(n—1)?—, and becomes infinite with n. Hence when x is large, any zero of ¢,(z) on the circle _z =n’, is at a large distance from every zero of ¥,(z). Thus the functions ¢,(z) and y,(z) have the properties mentioned at the beginning of the article. § 11. The functions considered in the last two articles are of finite order. As an example of a function of infinite order consider nD > sf F(z)= If Te = i |. where g>1, Ro >0, = [1+ leas ge which is an integral function when gq is an integer >2. Let p=p.i+ tp,, so that a, = (log n). It is easily seen that the conditions (a), (8), (vy), of § 7 are satisfied. Moreover, following the lines of § 8, we can prove rigorously that (6) (b) is satisfied; thus we can apply the general formula. The region R is practically* equivalent to the region consisting of the sum of all the annul / k (logs)? (a = Fi log ; , )< z\ < (log s)p (1 ~ |, where k' <4. s log s, The integer n is determined by log nx < 1/1 C,/2%. s=5 s=3 1 s= * i.e., by choosing k’ sufficiently small we can ensure general definition of R) sufficiently small we can ensure that the region R’ defined above is contained in R, while that R is contained in FR’. on the other hand, by choosing k (which occurs in the 46—2 342 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO un . ~ . Now % s?log,s, qua function of n, cannot be completely expressed, asymptotically or otherwise, in terms of known functions of n, Hence we cannot obtain a complete expansion for log F(z) in terms of z. We can, however, find an asymptotic expression of the form e47* [2 B,/r*). We shall neglect in P finite terms, and pure imaginaries which count as finite, for we propose to obtain an approximation for P in which we only retain the large terms above a certain order, We may then replace P by its real part P, ie. replace logz by log 7, and p by p;. pid We have | log 7 — p, logs n | =| pr (logyn + a — log, n) | < n log 7" Again S si = a ni4i+ K.n2, where K is finite for all values of n. ah pe 1 qu y mg ; Ky ) Sa o.8 Then B= a nii+a Ken ) (e, log. n +— logn pi = St log, s = fa ni? log, — p, 8%. logs s+ Kon? loge .....0..0.eseceeessasveees (1), where the K’s denote numbers which are finite for all values of n. x lien u(a)= | 8? log, sds. /3 Let t=exp (r), so that n+ 1>¢>n, and u(n+1)>u(t)>u(n). We have, Unta — Un < Unga < (nm + 1)2 log (n+ 1). Also u(n+1)>Xs?log.s>u(n). 3 Hence Dr log, s — u(t)|< u(n+1)—u(n), and hence is equal to K,.n%log.n...... (2). 3 Integrating by parts, we have AOE eesaal = ; y(t) «tae (3), where v(t) = ih (s%/log s) ds, From (1), (2) and (3) we have P = 50) + Ke. nt loge a sinerhs cide) Acieiteineetelis re ei Eee (4). Now v= 2 ertix = (on putting e*=~2), This integral, as is well known, possesses the asymptotic expansion*, va] 1 1 (i=) Ji | gta | ———___ 4. —__ a +———|], qgt+illogt q+1 (logt? qt1 (logt)' (log t) * The expansion follows at once by integrating by parts (J+1) times. FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 343 where Lt J, =0 t=x or, replacing t by exp (7), letayrlenb te Se aye = - 7 art eept +e case a eed Cee (5), where Lt J;= 0. Now log F(z) = P + (small terms) = P +(a finite expression) 1 q rk i v(t)+ K,ntlog,n, from (4+), Also Lt jew nt Pn? log, n] < Lt oe" log (rr) r? = 0. Hence changing / into +1 in (5), expliy+Iyrn} yy log F(z) = cca y eel PEs Soa aati »|. § 12. We shall not work out the conditions, analogous to (a), (8), (y), (8), of § 7, by which the general condition (A), (B), (C), of § 3 may be replaced, in the cases of other special forms of f(s), eg. e, [s?/logs]. When f(s) increases regularly and is of order greater than n**, where & is any number >0, and when the terms a, a... increase to infinity in a regular manner, however slowly, the general conditions will be satisfied. This will be the case, for example, when f(s) is of higher order than s? for all values of q. In this latter case it is always easy to find the dominant term* of P, and hence that of log F(z), as a function of r, but it is not always possible to find all the large terms of P, even qua functions of x. For example if f(s)=e", a,=s°, p=pit tps, and we easily find that the dominant term of P and log F(z) is ar o. €XP Lexp (7 — a)], where @ is the fractional part of 71>, To find P as a function of n, however, we should have to sum or asymptotically n n expand the series ¥ e* and ¥ logs.e, which is impossible by any known analysis. § 13. In §& 6, 9, 11, we gave examples of integral functions of zero, finite and infinite order respectively, for which the general formula of § 3 held. We can also obtain * By a ‘dominant term’ I mean a term Pj, such that P=(1+e,) P,, where Lt |e, |=0, rn=n 344 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO integral functions of transfinite order for which this formula holds. This can be done in two ways. We may take f(s)=s? (q an integer >2), and for a, a function which increases co very slowly with s, e.g. the inverse function $—'(s), where 3 (a) =e" ; where ¢;=¢.— .-5>18 and vy is the integral part of z. Again we may take a,=s°, and for f(s) a function which increases very rapidly &p with s, eg. f(s)=e,* , where e,=e,=...=an integer >2. Z Tis) In each of these cases the function II E +( ) | is of transfinite order. Us § 14. The functions for which the general formula holds have a certain common property. We have, when z is confined to the region R, log | F(z)|=P+R2C,/2". Now P= | S F(s) Jog r— | 3 F(s) log a ? s=1 s=1 and is clearly independent of the arguments of z and of the a’s. Hence on circles lying in the region R, whose centres are at the origin, and whose radii are large, | F(z) varies only by a factor of the form [1+e(z)], where Lt e(z)|/=0, r=n on the whole circumference of each circle. Hence, in particular, integral functions exist, of every type of order, zero, finite, infinite, and transfinite, which have this property. The property is characteristic of functions of zero order*, but quite exceptional in the case of finite order, and probably also in the cases of infinite and transfinite order. § 15. In cases where the general formula of § 3 does not apply, it may happen that the error committed by taking P as an approximation for log F(z), is finite. We shall give a theorem for this case for the particular form s? of 7 (s). This theorem applies to a more extended class of functions than we have hitherto considered: it applies to cases when the @’s do not tend to infinity, but tend to a definite finite limit. Suppose then that a, a,... is a sequence such that the moduli a, a... increase, and such that Lta,=a finite limit, which we may suppose without loss of generality to s=a0 be unity. When Lt a,;=1, andqg>1, it is possible for the product form ; reo=it[+ (2) to converge, and if q is in addition an integer, F(z) will be an uniform function. * Cf. a paper by myself, Proc. Lond. Math. Soc. Oct. 1907, §§ 4, 5. FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 345 Since F(z) has s? zeros on the circle z =a,, every point of the circle |z|=1 is a limiting point of zeros of F(z). Hence the circle z)=1 is a ring of essential singularities of F(z), the function F(z) only exists inside the circle, and the radius of convergence of the Taylor series which represents F(z) is unity. Let n be the integer such that a,< 2) < Gay. Then when z is near its limit 1, » is large, and hence, as we shall see, | P| is large. This fact enables us to apply, with a single reservation, the methods which we have so far employed, to find an approximation for log F(z) when |z| is nearly unity. The sole modification which appears arises from the fact that since |z| is finite, the \ 8) Fee formal development of log II [i+ (**) | cannot be an asymptotic expansion. This is the reason that we have not so far considered functions F(z) of the above form. In what follows we shall suppose that a, a... is a sequence such that a, is an increasing function of s, and such that Lt a, is either infinity or unity. In the first case s==s |z| is supposed large, and in the second case it is supposed less than unity, but near unity. In any case n is large, and the two cases can be considered together. We shall continue to use the following notation : Ip = | > ro) .logz— | S F(s) log a,| ; s=1 s=1 2 Zz \J(nts) R= Stog| 1 +( -) i ao Cn+s . n-1 ia / Qn ¢\F(-5) S= 5 log u + (“=1) | ; L } {2 J intl) ates [+(e]. and P=MHP, ete. § 16. Theorem. Let F(2)=1I E + tal iE where g>0, and where the a’s are subject to the following condition :— bir 4 (lleva cn oye] Se (3) S> WO) none ces noomnoncasonces sancecceene (1). s=a Let R’ be the region of the z plane defined as follows. Choose a positive number k which may be as small as we please and which we take in any case to be less than * This formula does not imply that the limit on the left-hand side is definite, but only that s?[loga,,, —loga,]>8—e« when s is sufficiently large. 346 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO both 48 and 7. When q is not an integer, R’ is defined to be the sum of all the wipe ifn Eee ic annuli a5 (1 Ta <| Z| < Os4s eee : When q is an integer, F(z) is uniform and has s? zeros on the circle z =4;. Let these zeros be ,b,, abe «++ qs: With centre ,b, describe a circle ,A, of radius a Then, in this case, R’ is defined to be the region of the 2 plane or of the interior of the unit circle (according as Lt a,=% or 1), which is exterior to all the circles ;A,, for all s=a possible values of s and t. Then when z is confined to R’, and n has its usual meaning we have log F(z)=P+Q, = n where = E silogr— = slog a | ; s = s=1 s=l1 and where |Q) m, 8% (log a4; — log a.) > 8’, s! a = , and hence Bet” Po L, sacasBivs cache oes eee (2). a, / We have, when n>m, s>2, 2) 7 qd q \@ ( Zz as E (Zen\n™ (jy (*a=) (n+s | > . = re, eee = i An+s On+s Anis On+s/ a (n+1) 4 a (nts—1)% n+1 n+s—1 An+s Onis << e76 CH) frome (2) lance suse cssceetlo-seenseceer ere cede e ee eeeeee (3). eee log (1 +2) aL Now when |2| | log E + a \< RS ete s=2 An+s s=2 K’ < i= Bani | ehowcseensvareeecreo teks ties a\e\e)s\einleleinioies leks (4). Next consider the term S. Assign a number @ between 0 and 1, both exclusive. | sd = 5 ; : Let w=) log E + (“) | , NV =[@n] (ie. the greatest integer contained in @n), and let m N m1 6,2 Sigs vop=t DS, Mon ee s=1 s=mt1 s=N+1 Then |S) . When m+11, we have n dx “9 Ge) sic r ue = fae [s-@ —n-W) _ Bs a = q—1 E < nN > B's E 2 ry pl D B's oe > gal (1 — 62) SAGs WHET Any o> Omeraccnerscctrse onan caecum oiseaientneties (He) If g=1, r da : , 7 ’ sip , = = 8's (log n — log s) > Q's. log (n/N) > §'s log (1/8) Si Nase WHET Ns os Omer nee hemes recente (ia) When ¢ <1, 31.8 iS dx 4 ie Gees a [ /n\}-9 ~ 1l=q ita) iS 1] es (ea Z 1| I=, \V Gey ip lye AF lia) cs 1| SOS, Willen: y=) Monconoaibacopenerasebaucadcc (7,). Hence in any case, mle 8 | = SS Nee wiheremAcs Ol ssn osedeeaisnsamensceeneen shee (7). Then from (6) and (7), we have (“) Iss COCO as Hop ORORCOSD OR OMe aDMnERGonCode (8), (As # vt | as S wr — Hence, as above, | log 1+ (“) < K |G) 32 (a) . (\" 2 JO S41 TT ik 12) NT (4s41 Aste a, ) J 0 LO-# siRvsssbrtabideecod whan ee (10) ; Cs s wire As - rus = =, Hence log 1 + (@) | HN f Ge (2) aK emt tia S) n—N-1| Gans (n—s)* and ot SS | log E + ( =) | s=] | & | n—-N-1 From (5), (9), and (10), S <0, +0,+ 05 < Kk’ a l=6>e Lae We have finally to consider 7, and 7). Suppose q is an integer, so that FR’ is the part of the z region which is exterior to all the circles ;A,. : . ka : : ‘ The radius of ,A, is ae z is exterior to all circles ,A,, and we have further |z|>a,. Let P be the point z, and let the centres of the circles ,A,, Let P’ be the point in which OP meets the circle |z| ap. Then Be ECE JU JAR oo5 IR We assume, what we shall presently show, that the circles ,A, do not intersect. Then P’ cannot lie inside more than one circle. If it lie inside any circle let it lie inside ,A,, and if it do not lie in any circle, let P, be the centre which is nearest to P’. Let ¢=POP,. TE UHAWE PP, PP,... PPS PP,...P' Pas | Tl P’ P| ED s=1 Ay... be) Bub vaneless Now by Cotes’ property of the circle | iI | / an? = 2. sin (An?d), s=1 oe ffi ppy ee sinket and hence |i PP,|, (P’P,a," >) = mae = : ‘ - 7 T Now since P, is the centre nearest to P’, ¢< 3 and hence $n?6< 5. n 2 FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 349 sin($n7p) _, sin(gn’g) 3 Henke sind a (jn%b) “sintd 1 >nt.—.1> n9/r. 7 ne a eal 2) EEN ht a Hence | aia | >—. | I] PP,| iC pee EeP,) Psi an s=1 Hey mor) gw a Ay, | k or — a An ere ni! kh Hence, since 2/2) an}, 1+ (*) ee AbaGmaroncoLiesoteenonocneacecd (13). The result (13) follows more easily in the case when gq is not an integer, when the region R’ consists of the annuli a, fee By _ a = Gd —e-s) ‘i when s>m, ae B’\ when s is sufficiently | Sra +e) (E). when s is sufficiently large, 2Q(1+e)k Seer === B Since <4, we can choose € and #’ so that 2 1. Then when s is sufficiently large et and hence the circles ;A, and +Ag4, S+1 — As cannot intersect. Hence the region R’ exists when gq is an integer. When q is not an integer it is again easily seen that A’ exists. In the case when Lt 4,=1, so that the range of z is the interior of the unit circle, s=a the area of the region &’ bears to that of the unit circle a ratio (1—A) where K FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 351 depends on k, and where K may be made as small as we please by choosing & sufficiently small. When Lt a,=«, so that z ranges over the whole plane, if we choose a large circle of radius p, the ratio of the area of the part of R’ contained in this circle, to the area of the whole circle, is [1— K(k, p)], where, by choosing & sufficiently small, we can ensure that Lt K (kh, p)1 is chiefly interesting when we have Lt a,=1, for, as we have said in § 8, when Lta,=2% and q>I1, we can _ usually apply the general formula of § 3. ia Consider the function F(z) = Il {1 + {z .exp (ws— 2) 7), s=1 where g>1, and @ is real and positive. This is a uniform function whose Taylor series has a finite radius of convergence, when q is an integer >2. When g>1 and is not integral, /(z) is multiform. The condition of § 16 is satisfied. In this case the most convenient function of 7, which tends to » as r tends to 1, in terms of which we can express the large terms of P, is Eacar We find that n=t— a, where 1 @ q-l + lata Calculating P in terms of a ry’ and neglecting finite terms we obtain 2 So ele ee fe be) = 5° G aa rl eecle i] ae where |\Q|< K. IV. Consider F(z) = Il E + exp (°) a" q>2, positive. s=1 The condition of § 16 is satisfied. We obtain, when z is confined to R’, ott a an ee Zz) = = | —~- 4a (1—2).o0%. | ——— rms ; log F(z) Ga) Feaal +4a(1—2).0! E Ta + terms of lower order, ee iain : 3 5 ena where* a=the fractional part of EaCp): § 19. In the class of functions for which the theorem of § 16 holds, and for which the general formula of § 3 does not hold, there are two types of functions for which we can obtain the finite term, as well as the large terms, of log F(z). = fli, ae The first case occurs when Lt a,= 0, g<1, and when G = ) tends to a definite non- s+1 zero limit, whose modulus is less than unity, when s tends to infinity. The finite terms of log F(z) can then usually be expressed by the use of the function my s=1 é s=0 xes” * In the last article of this paper we shall consider the reason why terms like a appear in Case IV. and not in Cases I. to III. FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 353 The general case is somewhat complex, and we shall content ourselves with finding the finite term of log [1 +(5)'], whose large terms are given in the last article, in the case when p and z are real and positive. We suppose z confined to the region R’. In the first place we must retain the finite terms arising from P. We find Pall etr— slog e—4+40(—1)+ 2a —a'| + (2) where a is the fractional part of 2°, and where Lt e(z) =0. We have n=2'//—a, z=(n+a). yee We must consider the terms 7,+ R and 7,+N. a) 1/, (n+s) n \ e(n+s) Wehare %+R— 2 [14( So)" [=e [as (eee | (. en n+s = n+s/ We proceed to show that \(2, + R)- S log (1 + er(-8)) | tends to zero when | tends to infinity. n + q\eluts) = es = p(a—s)]. Let Us = log E ~ (| = 2) | log [1 +e ] Choose a small positive 7. nea p(n+s) ; : S$ —2) (~=") = exp E (n + 8) log (1 one | When s | u,|< .B.« < An’, where A is finite. s=1 ere — I Choose 7 so that An* (=) : n+a 2n n < exp & pi. in” 4)* (when Iyn-? < 3) Again, when s>7.n < exp(— in. 7). 1 =D) n+a p (n+8) : a log i+ (*%) |+\loga ters #) Then when n+2a>s>ynn Us| < <2 exp (—4m!) + 2 exp [p(a—n!)]* [w+2a] and = |us|< (a+ 2a) {2 exp (— dyn?) +2 exp [p (a— nn®)]} s=[nnt aril Lastly, when s>n + 2a, n+ a\P (n+8) < p (n-+s) n+s/ (@) i ~o a ( n+a\P (n+s) s and S| te eS los ra ) [+= |deg (1 + ertesm))| s=[n+2a]+1 s=in+20}+1 | nae ° < 1 paps <4 Dpnts Jue s=nL-P 1, and Lta,=2, we have seen in § 8 that the complete asymptotic expansion of § 3 usually holds. When (1) is satisfied, g <1, and Lt 2,=%, we can prove the following formula, when z s=aD is restricted to le in R’: log F(z)=P+e(z), where Lt | (z)|=0. r=2 When (1) is satisfied, and Lt a,=1, we can prove the following formula, when z is restricted to lie in R’: » log F(z) =P + log II [1 + a,8*. e~ 879] + €(z), s=1 where Lt | e(z)|=0. T=] We shall omit the proofs of these formulae, merely remarking that in the first case we prove that |R+7,| and |S+7,| tend to zero with 1/r, and in the second case that |R+7,| and (S+7.)- > log [1 + a,8*.e-78]| tend to zero as r tends to 1. s=1 § 21. When Lt a,=1, and a, increases slowly to its limit 1, it may happen that ofits) does not converge for any value of g. We are thus led to consider functions defined by F()=1 [1+(2)"]. s=l ts / a product form of the type where Lt f(s).s~?=0c0 for all values of q, and Lt a4,=1. s=n s=n For functions of this type the following theorem applies. If Lt = Ra Tier) Sy SA) eae ae ere (1), then, when z is restricted to lie in the region R, defined (as in § 3) as the sum of all the annuli Gi’ dae log [1+ 4,7. eF 8], s=1 s= and Lt | e(z)|=0. r=1 Choose 8’< 8. Then when s is greater than a finite oc, we have, from the condition (1), T (s) ( = ) log [1+ ) ] s=1 | Ante <2 exp [—48’ (n+ s)], when n is sufficiently large, s=1 Next consider the expression S+ 7), Let v be a large integer independent of n, which we shall presently choose, and which we suppose for the present to be greater than oc. Then, when n>s>v, we have As Fs) / As J is) as Ts) oe fi 1 3 (*) < (ea =) < = log E + (“) | ee > log{1 +a,/ caro} | \s=1 s=1 < Ke" 4 K!e8¥ 4K” e+ y(v)(1 —|2)) <€. Hence the left-hand side tends to zero as r tends to 1, and we have the formula of the theorem. We notice that R-+7,) tends to zero as r tends to unity, as is seen at once from (2). ao As an example consider F(z)= II [1+{z.exp (e~*)}*“], which is an uniform function s=1 if e* is an integer. When z is restricted to the region R we have 1 bene ack Qe-aa 1 log F(z) = 53 - ee) oma rad () 0 : log 5 -da —a@ (1+a) = ad — ae ae e ke a) T-e** dey +7 et (t—a) _ ee acca oe en l-—e" (em) 2ea i yens)2 + e(z), 1 8 log — te} Tr 1 . . . ca us. where t= = log a 6 =arg (z), a is the fractional part of t, T= = log [1+e-*.e-'], and where Lt |e(z)|=0. r=1 The terms in crooked brackets count as finite terms. § 22. In the case of all the types of functions we have so far considered, we have always had the inequality |R+7,+S+7,|< K,so that the expression P represents log F(z) with an error of an order at most finite. It may happen in the case of certain functions defined by product forms of the type we are considering, that |Q|=|R+7,+S+T7,) tends to infinity as r approaches its limit 0» or 1, but is of less order than P. If we then can calculate P as a function of r, with an error which is of the order of | Qj, obtaining P(r) say, then we have log F(z)=P,(r)+ terms of the order of |Q), and P,(7) gives an approximation for log F(z). We shall not develop a theory of this kind in detail. Any particular case can usually be worked out quite easily, and we shall content ourselves with a single example. 48—2 358 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO Consider the integral function 2 zZ {s1/?) rom i li+(3)"], where p=pi+ tps, p; > 0, and [x] is the greatest integer contained in a, For the present we suppose z unrestricted. The integer x is determined by n+1>r°">n, so that n=r'"—a, where 0n+1>n>v% Then v+1>t>v, where t=r'™, and v=t—B, 04s, n+s<2n), where X is finite. < exp [- K.n- It follows easily that ue < | log (Kin ™”.s)|.exp[— Kn “".8], where 29>[g'7)....2:.0-.00coeeeeee (2). From (1) it follows that [nr] = u 4). eriae is 1 — exp (— Kn’) () Finally when s >, (2 + Med 2) l 2/2 n+ S$ S 2 / Dae n —\sl/2 . *.° u;< 2 > e™ s=nt1 s=n+l1 ie 1/2 < 2 | e =" dx 0 (a) = 4? + 872+ (@. term: of order p)/,.53...06..0se0eccceverens (8), s=l1 and y [s""]. logs = = m[log m? + log (m? + 1) +... + log ((m+1)?—-1}] s=1 m=1 = Shim log [T° {(m + 1)}/T (m?)] 1 — which reduces, by the help of Stirling’s theorem, to 4y¥logy—4v3+ 3 logvy+4v?+(a term of order v) ........ceceeeee een ee (9). Substituting from (8) and (9) in the expression for P, then substituting »=t—, expanding the terms in py in descending powers of ¢, and arranging, we finally obtain P=4p,r°"' — hor + Mir where | | is finite. Hence when 2 is confined to the region Wt detined above, we have log F(z) =p. re tPi- gee eN bee, where | V| is finite. 360 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO § 23. We have so far only considered functions for which the index f(s) tends to infinity with s. Our methods may also be applied to approximate to log F(z) in the cases when f(s) is a constant, which may, without loss of generality, be taken to be unity, and when f(s) tends to zero with 1/s. I have shown in another paper* that when /(s)=1, and when a,=s*®, where Lt $(s)=«, and $(s) is a non-decreasing function of s, the expression P is of order s=00 not less than n(n), and |R+7,+S+7,) is of order not greater than the greater of kK ae Thus P is an approximation to log #'(z) with an error of order K or p(n) Fay" This theorem applies to all integral functions of zero order, the modulus of whose sth zero increases steadily with s. We shall then not consider the case f(s)=1, but shall consider the case when /(s) tends to zero with 1/s. 2 conn Z F (8) -) gi (8) We rewrite the form II Fee) | as II (ee | s=1 Cs s=1 L bs where b; =a,/". The function #’(z) is clearly multiform, and must be made uniform by a cross-cut from the origin. Theorem. fla S re Let F(e)= Il ji+(2) [=a Eee e s=1 Cs s=1 bs where |b;|= 8s, and where 6,’ =a, is an increasing function of s. Suppose the following conditions are satisfied: il Lt =~ =o. ee) (8) Gh is a non-decreasing function of s after a finite value o of s. Gye Tee s=a $+1 Then when z is confined to a certain region #’, we have the formula log F(z) = PaO. where P= [ > J (s)). log r — = log Bs, Sel s=1 n is the integer such that cote | A Sace. and where |Q| 2, | gf (nts) pt (nts) Onis | Bats pi in) Se < re from condition (8), ns S rt (ny [oper ne Bnss1 Bien [efewl Briss UN Sal ng wee saccades nc neaennceveatasesaes ciesccawiee accmecence 6 aie Gu) ON ae cata ora w cae tema He ace Ca tia tioeee ee Sone ow Se ee iaks oOsacaeieee (2) | gt (nts) gf (nts) | : Hence |log [1+ | .| | log E Ee | s=2) n+s !} <5 >i s=2 Vi : whence, by reasoning similar to the above, S ‘| 1 bs finite K. =< Oo a nite 2 s=mt1 /°8 | 2 rss) s Also when n>, m | e . at - > | log E + 75 | is finite and < K,. 1| (ge ‘ m b, | Be | bs Hence Ss ay log E 35 Hs | es log E i =a | Zeit Vabinines J 267 adesonoccscoccO BOCs UGnee aL HOOH UO CFE COUR EESED Ee OCUcCCeuCCORreC (4) We now define the region R” as follows. We choose a positive », which may be as near to 1 as we please, but which must in any case be greater than na. 362 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO Then R” is defined as the sum of all the annuli 1 1 PTD. gay S21 eG, een onecnesccede ctecssaent ete sees (5). We have 1 VF (st) ——— ' S(s+l - BION O44 aie A aa er s=a = s=D 2s ris as > Lt Eee since f(s +1) > f(s), s=” B > (ur) Sil It follows that and therefore that the inequalities (5) are possible, so that R” exists. Now when 2z is confined to R’, gf (nH) b,, air and Fe | Dns are less than a positive constant which is less than 1. If follows at once that P| and): |/'7'|\ "are Gites... .sese cess sess cerees eee sen (6). Now we have log F(z)= P+ey2+ R+7,4+84+T7,, where |y.|\ <7 From (8), (4), (6), and (7), we have log F(z)= P+Q, where |Q)0, @=@,+ la, @,> 0. s=1 Calculating P in terms of 7, we find : ! le Cid 24 When g<1, log F(z)= ere w, 1*4,(logr)1*+7 + €(q).logr+(finite terms). When g=1, log F(z)=4 log log 7. logr+[y — 4 log (ew,)] log 7 + (finite terms), where y is Euler’s constant. Ee ae .@,271 , (log r)4+1 + (finite terms), When g>1, log F(z)= €(q). log ee Thus when g>1, | #(z) behaves asymptotically like 7$@, FUNCTIONS DEFINED BY HIGHLY-CONVERGENT PRODUCT-FORMS. 363 § 24. The methods which we have used may be applied to obtain approximations to other functions than those we have considered. For instance, in the theorem of § 16, we may substitute a function like s?(logs)” (log, s)”...(log,,s)% for s%, wherever the latter expression occurs in the statement of the theorem and in the condition (1), In general our methods may be used to approximate to the function ca z. gs) )x®) F()= I r+ P@2 Boh = ls where (2, s) is such that Lt kz ot §)| =0, for all values of z, where ¢ is such that there is always a single n, such that eke) >1, and jegutesenipe 2) < An, On ile and where the product form converges sutticiently rapidly. The latter condition will generally be ensured, @ and y being supposed given, when |a,| imereases sufficiently rapidly with s. The approximation which we obtain is P= [x (s) (log $(z, s)— log a,)], s=1 and the general rule is as follows: When the product form converges as rapidly as or more rapidly than I (1 +5), s=1 RP is an approximation for log F(z) with a finite error, and when the product converges as rapidly as, or more rapidly than some integral function of zero order, WP is of higher order in n or r than [log F(z)— RP], and RP is an approximation for log #(z) down to a certain order. In certain cases we may possibly obtain some more extended formula analogous to that of § 3. We shall not develop this theory further, but shall now show how we may approximate to certaim products which are of a different type to those which we have so far considered. § 25. We shall consider functions of the following type: F(2)=IL [1 + u,ef], sh where Lt ‘ ia) where | ws|<1 for all values of s, and where > w, is divergent. s=1 In the case where f(s+1)=m,f(s), where m; is an integer which becomes infinite with s, F(z) is an uniform function, and the radius of convergence of its Taylor series is evidently unity. It can be shown, moreover, by a proof similar to the well-known Vou. XX. No. XIV. 49 364 Mr LITTLEWOOD, ON THE ASYMPTOTIC APPROXIMATION TO proof in the analogous case of the Taylor series* La,z™™-~™, that the circle |z|=1 is a ring of essential singularities of #’(z), and that the function consequently exists only inside the circle. Since |u,;!<1 the function has no zeros inside the circle, and is of a completely different type to that of the functions we have been considering. Let us consider the general case when Lt —~— =o, and z is real and positive, less than unity, but near unity. There are two cases in which we may obtain an approximation, I. Suppose that for each value of s, one or each of the following conditions is satisfied : (a) |u\<8<1,; (6) |argu,|1. Then when s is greater than a finite p, m,>h. Hence when n>yp, s> 2, 2f(n+8) = exp ((log {z7™™)}] . ang. Mnto +--+ Mn+s) < EXP (— Mpy--- Mnys), by the definition of x, < exp (— h*) s> yp, 1> 2/8 > exp [log {2/™} . /(my Mn»... Ms)] >exp (- #3 -} Wr cee Vln > exp (—h7"*). Hence [1-2] <1—exp(—h-“""="") 5) AON SIN | race vac [ tt(1 — 27) eS 1+ Us || | log [1- * a =| » from (1), j1-™ - mend | 4 ee =.h-™=>), provided h ar c Hence, when n< s< yp, log | log (1 + ws) — log [1 + ty 2 ‘\= logd+u)— = log(l+u,zf)]|<= = h-™* CSE M) Sy, nie 12 Gy, coodoo0ocer (7). s=1 s=1 } Finally, it follows at once from (a) or (0), and the fact that 2/™ and 2’) are positive and less than unity, that log [1 + m2? ]\+- log [1 +e, 27") leg (l— 27] Sa finite KG ee ..ce cos (1). s=1 Let 2/™ =e-*®, so that 0<@2q, ¢.s! is a multiple of 27 and 2! is real and positive. The first 2¢ factors of F(z) only give a finite factor in the approxima- tion, and we obtain the same approximation, viz. Il [1 +1] =2", as when z is real and positive. pe The integer n is determined by |jzji=e" O<0 \ \ | ——_—_,,, k=1 1. $15 exp [(a@+ sw)*]’ | | ap lcea*) | R| tends to infinity | | Only known functions occur I. $16 ; | B ie | | IL. § 22 sP ) ) P pie ) ) I. $12 my | | | | | Abs PSs | | | L $14 (a + vs)Kese | | z sf He 8 The large terms contain only known E aS, »ql nist | IT. § 23 ese? Gea 0 | ‘ ) Esti os }) \ $16 exp (¢*) | | ae LEE I. $15 exp [(a@+sw)*]’ fs | [ g i | . Terms in a occur in the large terms Il. $6 exp (e*) 5 Zz /82 4 [=] »q>l f | &| tends to zero | II.§9 s@ | 2exp] »q>2 | | II. § 18 > - || IL. § 21 [z exp (Gayle ) Za alt P requires new functions for its IL $11 (log =A end } complete expression ; XV. The Reality of the Roots of certain Transcendental Equations occurring in the theory of Integral Equations. By H. Bateman, M.A., Fellow of Trinity College. [Received May 4, 1907. Received in revised form October 16, 1907.] § 1. In a former paper* I showed that if ¢,(s) is the solution of the integral equation F()= G0) -r] 6 LMBN(OAG raoaicownecionsma case teen aac. (1), the coefficients in the expansion of an arbitrary function y(s) in a series of the form x (s)= Tada (s) pe cccecedecccvercnssreessccrcerscveceseseeces (2), where the summation extends over all values of X for which ¢,(s) satisfies a given linear condition br (*) =, are given by the formula [xe t) x (t) dt é te ay Pa(*) Or where K(s,t) is the solving function of the integral equation and F(«) is a convenient symbol for the result of imposing a linear condition of a distributive nature upon the function F'(s). The results obtained, however, were in some respects incomplete because the roots of the equation pa(*)=e are not necessarily real and there is nothing to show that they are infinite in number. The latter condition is necessary if the expansion (2) is to be used to represent a very general type of function. The object of this paper is to carry the analysis a step further. The particular case that is considered still leads to a very general class of expansions which is seen to include many of the expansions which occur in the boundary problems of mathematical physics. * Camb, Phil, Trans., Vol. xx. No. xu. Vor, XX. No. XV. 50 372 Mr BATEMAN, ON THE REALITY OF THE ROOTS OF The particular condition that is chosen is ik f (8) $2) Gace eee: ee (4), and we limit ourselves to the case im which «(s,¢) is a real symmetric function of s and ¢ which is continuous for a i <4 >) and y(t) is continuous in the interval 0. It will in general be infinite at the singular values of » and so will take every real value once and once only in passing from one infinity to the next*. rb ph Next, let QA)= | K (s, t)«(s) #(t) ds dt. b The formula = LAM) = [ K (s, x) K (a, t) dz, which is deduced from 0 Oe a da (Ss) =| K (s, 2) $y (x) da, by putting /(s)=«(s, t), da (s)=K (s, t), enables us to calculate the rate of increase of 0()). We have in fact 0 rb vb som=[ [Ako t) aw (s).c(t) ds dt =f [KG r) K (r, t)a(s) # (t) drdsdt. Now K(s, t) like «(s, t) is a symmetric function of s and ¢ and so the triple integral on the right-hand side may be written 7) | bemrar, a b where x (7) =| K (r, t) x(t) dt. ~ a * If 5(d) is the determinant of the integral equation the _it is easy to verify that the solving function is : y relation if ; te & Eee *n) eae | K (rr) a Hin j=— t, Yrs Und 5 (A) J a 2 a(F : l : Yr + Y, shows that the function — {log 6 (A)} decreases as ) in- ; 7 : dn Now when «,=y;,... %,=Yn- the function creases. are ’ (8) Uy --- Uy Again, if with Fredholm’s notation hes ONG t,o. (s, t)= ae 5 | kK (8, t), K (8) Y)s «+9 (8, Yn) | He = Boson ee | x (ay, t); «(25 9a)s | es EN | Sod xSa OA ESO OSES IC DORIOOOII 12 is a symmetric function of s and t. The theorem that | x (an, t)s ~ k (aq Yn) | H(s, s) increases with thus tells us that the function we form an integral equation K (3 Eee: a) By ayes on ‘(s\=a(s cents eae | F()=G(s)-—A] hls, ) g(t) at, K ( a Sal Up a adie =) increases with \. The roots of K iS a ea =| are 5 : ft, U1 -°s Un $, %... Ly in which h(s, j= —— ; 2 . K ies 3 -) therefore separated by those of K( soy ")=0. Trios ds cease CERTAIN TRANSCENDENTAL EQUATIONS. 379 Also, x (7) cannot be identically zero for all values of r in the interval (a,b) unless a function z(t) exists for which o=[" e(r, t) w(t) dt, for this equation is obtained by putting »=0. b The quantity i [x (r)P dr is thus positive and so £0 is positive, this implies that © (X) increases continually with 2. If Q(A) is zero for \=—o this theorem can be used to determine a range of values for which the double integral is positive and consequently for which K(s, t) is a definite function. () will clearly be positive until the first singular value of A is reached. The values of X for which ©(X) takes an assigned value are separated by the singular values of X. This fact is of some significance in connection with the problem of representing an arbitrary function F(X) in the form b pb Fa)=| / Q(A; 8, t)x(s) x(t) ds dt, “ava where a(s) is a continuous function*. Thus in the particular case when Q(A; s, ¢) has the form K (s,t) the equation can only possess a solution if F(X) increases continually with A. § 3. We next inquire whether any information can be obtained with regard to the values of X for which the homogeneous integral equation + b 0=| [76 )-r96s, Nd at, can be satisfied. Let us suppose that f(s,¢) and g(s,¢) are real, continuous, symmetric functions of s and ¢ for (a» =0, in other words, the singular values of X are all real, If g(s, t) is also a definite function the singular values of \ are seen to be all positive. § 4. It is known that when «(s,¢) is a real symmetric function, the values of » for which the homogeneous integral equation wD G()=—>| (6, BGC) dE =O eceerseeseener ase (1), a can be satisfied are all real. When «(s,¢) is not a symmetric function the singular values of % may be complex quantities and some assumption as to the character of «(s, t) must be made if further information is required, Let us suppose in the first place that (S50) =1—V00 (ty S) cis seia'e Sa asdedisleceueinea asissesce cee meetee (2), we have then oT) p(t)+r] (7, t)d(r)dr=0, rb pb hence C0) (s)+ | x (s, t) «(7, t) d(r) drdt=0. b Now i «(s, t)«(7, t)dt is a symmetric function of 7 and ¢, consequently ? and $(s) a are real. Further, the above equation gives b b rb pb | g(s)ds+r{ [(s, t) «(7, t) hb (s) $ (7) drdsdt = 0, rh rb or | g(s)ds+r2[ ye) dt=0, rb ? where nV @)= | « (s, t) f (s) ds. This equation shows that \? is negative, consequently the singular values of % are purely imaginary quantities*. * This is simply an extension of Weierstrass’s theorem for a bilinear form, CERTAIN TRANSCENDENTAL EQUATIONS. 381 Next, consider the case in which ACU) = [rs DG) (i VUE SERRE peRB abn Oaccocacbnco pb neetnnceee (3), f(s, «) and g(a, t) being definite functions. It is clear that if rb (=| AG OVEL ENE OE ERR PER (4), “@ b the function ap (x)= / tH) (ay QVC OOD. weno pcOOoB OCC BREE CS OnRnE EASE Tr ETOAC (0); is a solution of the adjoint homogeneous equation “b Pe) Ma A Cline 1) Sonny denna nvasesdedet buessavse: (6), for we have f 6 rh pb rh : (7) =| g(r, t)d(t)dt= »| t [ g(r, 8) f(s, 2) g (a, t) $ (t) dsdadt = mE g (7, 8s) f(s, 2) W(x) dsda =X is ab (x) « (x, r) da. Now let NEA tA. DiS) ONS) ctr Omi (8) Meee ea fo voz dasonta oe oe eee nen nee (a); (8) = vals) + tYr(s)! then a= i Mie ook, a, (8), rh wp(a) = I CR) OO CC (9). Also since o(s)= af « (s, t) db (t) dt "hb fb =5| i F (8, 2) 9 (a, t) b(t) dedt, b we have o(s)=Xr [ JAG BAP (@yCacctusedbaoskconecer Ree eeacsooeobE (10). Equating real and imaginary parts we get rb rb g; (s) = »| T(s, 2) Wn (x) dx — r.| FT (S, &) Wo (ax) da | rb rb od, (s) = »| SF (8, ©) W(x) dx + rz | F(s, 2) Wy (a) dx Now it follows from (8) and (9) that a ie [do (@) Wri (x) — o; (x) Ws (x) ] dx =0, 382 Mr BATEMAN, ON CERTAIN TRANSCENDENTAL EQUATIONS. hence if we multiply the first of equations (11) by w.(s), the second by y, (s), integrate and subtract, we obtain 0= rf [fo x) [Wri (8) Wu (@) + Wo (8) Yo (x) ] dsda. The double integral cannot be zero since f(s, «) is a definite function, consequently A, is zero, that is X is real. Again, the equations yy (@) =[" 90, t) p (t) dt, b gar] fo) Vode, give ae g (a, t) b (x) b(t) dadt =| (x) (x) dx = rf [re x) W(s) (a) dsda. The double integrals are both positive, and so X is positive. Thus we have the theorem If AGS =| Fe x). 9 (a, t) den where f(s, x) and g(a, t) are definite functions the singular values of X for the homogeneous integral equation o(s)—r] e(s,) &)at=0, are all real and positive. XVI. On the Solutions of Ordinary Linear Differential Equations having Doubly-Periodic Coefficients. By J. Mercer, B.A., Trinity College, Cambridge. [ Received, November 13, 1907. Read, November 25, 1907] INTRODUCTION. The systematic study of linear differential equations, having their coefficients doubly- periodic and solutions all uniform, owes its inception to Picard, who, in a note to Comptes Rendus* shewed that, in general, the integrals of such equations are linear combinations of doubly-periodic functions of the second species. A subsequent note from Mittag-Leffler+ pointed out the theorem, now known by Picard’s name?, to the effect that in all cases there is at least one solution which is a doubly-periodic function of the second species. Neither of the writers however gave an account§ of the effect which multiple roots of the fundamental systems of period equations have on the analytic form of the integrals. This remained for Floquet!) who, by means of an induction, is able to establish that in the most general case the integrals are linear functions of polynomials in u and » (I, § 35) having coefficients which are doubly-periodic of the second species. But the investigations of this writer have the defect that no precise information as to the degrees of these polynomials in uw and v is attained: we are never sure how many of the coefficients are evanescent. Since Floquet many writers have dealt with the subject, notably Jordan{, Picard**, Schlesinger++ and Forsythii{; but nowhere is the point just mentioned made definite. In section J. of the present memoir we attempt to make good the omission just noticed. The method used may best be described as the two dimensional analogue of that which would have to be employed for the case of a single period, supposing the VEc. Norm. Sup., u1.™ séries, t. 1. (1884), pp. 181—23s. {| Cours d@’ Analyse, t. 11. * t. xc. (1880), pp. 128—131. + Comptes Rendus, t. xc. pp. 299, 300. + This is only just since Picard in a previous note to the same publication (pp. 293—295) shews it to be true for equations of the second order. § We refer to equations whose order is general. Picard in the place just quoted considers equations of the second order. || Comptes Rendus, t. xcvm. (1884), pp. 82—85; Ann de Vou. XX. No. XVI. ** Traité d Analyse, t. m1. (1896), pp. 406—417. tt Theorie der Linearen Differentialgleichungen, u. 2 (1898), pp. 403—424. ++ Theory of Differential equations, Vol. rv. (1902), pp. 441477. In addition to the writers quoted mention may be made of a paper by Stenberg, Acta Mathematica, b. 15, pp. 259—278. 52 384 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR theory of elementary divisors unknown. There are however two directions in which we make the problem more general than that considered by previous writers. In the first place we do not limit ourselves to the case in which the coefficients of our equation are mere elliptic functions. We adopt the wider condition that they should have for their singularities a reducible set of points, and then, thanks to the theorem quoted in I, §1, we are able to obtain our results with no more difficulty than when their number is finite. The second generalisation follows as a result of our investigations in §§ 1—5, where we find that the relation aba +b = 1, on which depends the theory of the equations studied by Picard, is true of a much more general class. The only drawback is that we are limited to the consideration of particular branches of solutions, localised in a doubly-periodie region (§ 7) and are therefore unable to define precisely the nature of the remaining ones. The defect, just mentioned, being kept in mind, section II. is devoted to the consideration of a very particular class of solutions—exclusive solutions as we call them (§ 8). A detailed discussion of Halphen equations (§ 10) is given, and as a result of it we are able to see that Halphen’s conclusions in respect to the particular case are still true of our more general one: we are able however to state our results in a somewhat neater form, by the aid of the artifice of § 12. Finally we consider the most general possible exclusive solution these equations can have; finding that it must be a solution of one of Halphen type (§ 22). We may observe in conclusion that the results of section II. justify in some measure the more general point of view adopted in section I.: in effect the solutions dealt with are integrals of equations of the type there discussed. I. § 1. Suppose that we have a linear differential equation of order m, say d™w d”™w d”™—w , ae =p (z) dz = fotatare + py (Z) az oa + Dm (2); where pp», -+» Pm are doubly-periodic functions (common periods @, w’) which are (1) uniform and (2) such that their singularities are a reducible set of poimts: in the sequel we will describe functions of this type merely as uniform doubly-periodic* functions. Instead of restricting ourselves to the case in which the solutions are all uniform, it will at least be instructive if, in the first imstance, we consider the most general kind of solutions possible. Take therefore any base point z, which is not a singularity of any of the coefficients p,, Ps, --- Pm. At this point there will ‘exist m linearly independent power series in (z— 2) which are solutions of the equation; and the radius of convergence of each will be ¢|c—z2,| where “c” is that singularity of the p’s which is nearest to 2. Let us call these Bi = Zo); Es (z— 2); ose fer (Zz = 2); * These functions include elliptic functions as a where @ is the usual Weierstrassian elliptic function, are particular case: the functions @(@ (u)), @(@(@u)) ete., doubly-periodie but not elliptic. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 385 where 120 (C= 2 = Sn (BAe Any one of them P,(z—z,) together with its continuations will constitute a certain monogenic * function (f,) whose singularities are among those of the coefficients of the equation. As the latter consist of m reducible sets they are themselves a reducible set (S); and so the points of the plane, other than these, form a completely open region or domain+ (Il): obviously (IL) is doubly-periodic?. In general the set of singularities of each function f, coincides with (S); but in particular cases it may happen that the points of a certain component of (S) are apparent singularities, so far as regards (/,); the set will then consist of those points of (S) which do not belong to this component. It is a fundamental property of monogenic functions that if @,(z¢—2) be an element of (f,) whose base is A, that is to say an element derived by continuation from P,(2— 2), then it also satisfies§ the equation. This property of (f,) is expressed shortly in the statement that (f,) is a solution of the equation. § 2. Let us now fix on a definite path (y) from 2 to z+, which passes through no singularity of the coefficients; and let us continue the series P,(z— 2%), ... Pm (2— 0) |i along it so as to obtain new power series in 2—(2,+@) say P,.(2—(4+)|¥| 20) = > Ms brn? (z — (Z,+@))”. (r=1,2,...m)|c—4|>|z—(4+.0)| Each of these, as we have said, satisfies the differential equation of § 1. Thus, if we write z=w-+t, we see that a 2 Prn® (t fa Zo)" (7 = I. 2. eee m) 0 are solutions of the equation dw d”-w dt” SPP @) dt” +... + Pm (E+), 2 3 d™w d™w ie. of qe ~P (t) qe + 22. + Dm (t); since the coefficients are periodic. It follows therefore, replacing ¢ by 2, that each of the power series > pet (2—2%)" (say Q,(z—-%)) which have z for their base and a radius of 0 convergence ¢ |¢—2,| are solutions of our equation. * «“Monogenic function” is here used in precisely the same sense as ‘‘analytic function” in Harkness and Morley’s Introduction to the Theory of Analytic Functions (e.g. p. 154, § 90 of the 1898 edition). Cf. Baker, Proc. Lond. Math. Soc. 2nd series, Vol. tv. p. 116. + These words are used in the strict sense of Young, Sets of Points, Ch. 1x. The reader who is acquainted with the theory of regions will not find much difficulty in estab- lishing, by means of an induction, that if R is any region and f a reducible set of points internal to it, then the set of points (R—) which belong to R but not to 8, is a region. This theorem is a particular case of a more general one. + A region or domain is said to be doubly-periodic (periods w, w’) when it is such that if z) be a point of it Zp, Z)+w' are points of it. § At all points which are not singularities of the coefficients, i.e. not apparent singularities above-mentioned. In such a case the expression—in the strict sense at any rate—is meaningless. We always suppose this taken into account when speaking of a function as satisfying a differential equation. || This continuation is possible because (II) is a domain. 52—2 386 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR These series are all linearly independent. For if not we would have a relation of the form m oO | | XL, & pinY (2 — 2)" = 0, |c—%|>|2z—2| 1 0 where the /’s are certain constants which do not all vanish. Consequently we would have m wo SLE pnt (2 — (2) + @))” =, lc—2z|>|z—-(4+@)| iAneKO m or > 1,P, (2—(& + @)|y| 2) = 0. 1 Continuing the left-hand member of this equation back along the path (y) to 2* we deduce 3 1, P,(2— %) =, 1 which is contrary to our hypothesis that the series P(z—%) are linearly independent. It follows from the general theory that we have Q, (€7 ws Zo) = On P; (2 cm Z) at Ora Po (2 a Z) +... + Arm Pn (Z aa 2); @=1) 2a) where the a’s are constants and the determinant of the matrix Gi) Ghee sassoe Oamn ; Gian | GEN auscas (Oem (ra Gh odeboo Gem Gina Ome POG Omm called a+ matrix of the system for the period @ is not zero: symbolically this result may be expressed as Q(2—%) =aP (2— &) where (a) is the matrix just written. § 3. It is obvious that a similar result applies to the other period (o’), viz. if we take a definite path (y’) from z to 2+’ and continue along this, the series P, (2 — 2) becomes P, (2— (2 + @’) | y' | 2): (r=1, 2, ... m)|c—4|>|2z—(a+o) the latter gives rise to a power series oa > Pon? (2 = Z)", 0 * We can always suppose that our continuation along + Of course it is obvious that this matrix may, and (y) from 2) to (zj+) has been by a standard chain. in general will, vary with the path y. Harkness and Morley, op. cit. § 90, p. 156. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 387 say Q,’(z—2), which is a solution of the equation, and such that Or Z—H)=b,P, (2—%)+bPs(z—2)+ tee + Drm Pm (2=2); @r=1, 2, ... m) or briefly Q (2-4) =bP (2-2), where 6 is a matrix of the system for the period (@’). § 4. Consider now what happens when we continue the series P,(z—(z,+@)| | 2%) from (2,+@) to (%+@)+' along the path y” congruent* with 7. Since P; (2 = (2 35 @) a Zo), is obtained from Q,(z—2,), ie. from Ay, Py (2 — 25) + GaP» (2 — 2) + --. + OpmPm (2 — 2) Zo+0+0" by writing z—o for 2z, it is clear that P.(2-(4+o+0) y" 2+0!¥) 2) is obtained from An P, (z — (2 + @’) | 9 | 20) + GPs (z—(2 +0’) y' | %)+ DE I 5 ken Se io eens + Arm Pm (2 —(2) + 0’) | y’ | %), by the same substitution: Le. is obtained from An Qi’ (2 — 20) + Ae Qs’ (2 — 20) + --- + Arm Um’ (2 — 20), or from 3 Ars & be Pe (Zz — 2), (r=1, 2,....m) s=1 t=1 by the substitution of (¢—@—a’) for z. This result may be symbolically written P(z-({+o+o0) 7" 2+ ¥| %)=abP (z-(2+o0+0’)). In a similar manner if (y’) is the path congruent with (y) which joms (z,+o’) and (2) + @ +’) we prove that P(z-(+o+) 7 \a+o' \y %)=baP (z-(2,+0+0’)). Moreover, since P(z—%), continued to (z,—@) or (2—o’) along a path congruent with * i.e. y'” consists of points congruent with those of 7’. Morley, op. cit. p. 243. For the definition of congruent points vide Harkness and 388 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR y or vy’ becomes a*P(z—(a—@)) or b1P(z—(z,—o@’)) as the case may be, we shew similarly that P(z—(™+o')| yy |ato+o’ "| 4 +@|y| 4) = abaP (2 —(2,+ ’)), | 29+ @ | y| 2) = aba1b P (z— 2). P(z—4|7'|%+o'|y"|~+o+o'|%” § 5. From the equation just written it appears that if the integrals of our equation are all uniform we must have abab2=1; but the converse is not necessarily true. We proceed to find a necessary and sufficient condition that this relation between the period matrices a and b may hold. Imagine the pseudo-parallelogram, whose sides are y, y’, 9”, 7, shrunk in any manner so as not to pass over any singularity of the coefficients p,, po,... Pm; in this way the interior will become a certain region or set of points ©. Since the set (S) is closed © may be chosen so that the points external to it and congruent sets is a region: for instance, if the singularities in a parallelogram are finite in number, say ¢,,¢...¢, @ may be taken to be the lines joiming ¢,¢., ¢: Cy, ... Gc, provided they do not cross. In any case however this region is doubly-periodic: let us call it @. The necessary and sufficient condition above referred to is that the branches of fi, fs, ... fin, deduced by continuation from the series P,(z—%), -.» Pm (z—%) and considered as localised in ®, should be uniform. The condition is obviously sufficient; for by hypothesis the series P(z—2) when continued round y, y”, y”, y returns unchanged, ie. by the last equation of the last paragraph io = il, As to the necessity of the condition, consider any closed path T passing through 2, and lying within ®. Without passing over any of the points of S, this may be deformed into a path consisting of, firstly a path joiming z to 2+ p,@, which consists of | p,| paths congruent with y; then a path joining 2+p,@ to (%+p,©)+q@' consisting of |q,| paths congruent with y'; then a path joining 2+ p,o@+qmo and (4+p,o+Mo)+p.@ consisting of |p| paths congruent with y; and so on. Thus the series P(z—2) when continued round I’ become aP: 6% aP: b& ... as b%s P (z — %), & s where > p,=>q,=0, since the path is closed. 1 1 Supposing now that Cte Oma we have of course ab=ba. As matrices obey the associative law it is easily shewn from this that a product of Xa’s and wb’s written in any order is equal to ab*, Thus the series P(z—2) when continued round [’ become 8 8s > > Pr =r ae bl P (z—%), i.e. become P(z—%). Thus the condition stated above is at once necessary and sufficient. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 389 Equations which possess this property will in future be called generalised Picard equations: they include ordinary Picard equations, i.e. those whose integrals are all uniform, as a particular case. § 6. It is an easy matter to express the condition aba} = 1, in terms of the behaviour of the series P(z—2,) when continued round the various singularities of the coefficients when the number of the latter contained in a given period parallelogram is finite. For we have seen that when they are continued round a closed path enclosing ©, the series P(z—2) become abab7P. But this path may be shrunk so that it becomes a series of loops enclosing successively the various singularities Ci, Co... contained in ©. Consequently, if the series P(z—2z,) become m,P(z— 2%) after continuation round the loop enclosing c,, they will become MyM... MP (z — %), after describing the series of loops just mentioned. Thus we have abab = mm, ... Mz. The condition that the equation should be one of generalised Picard type is therefore expressed by HONE nao Wn Ile In particular, if there is only one singularity ¢, we must have nn ie. in the case of a single singularity the only generalised Picard equations are those for which the integrals are all uniform functions, i.e. those of the ordinary Picard type. § 7. Suppose now that our equation is of the generalised Picard type and that ® is the region within which the branches of f,, fo,...fm are uniform: ® may be called a doubly-periodic region for the equation or simply a doubly-periodic region. Denote by 9: (2), Go(Z); ++» Jm(Z), those branches which are deduced from P, (z— 4%), ...Pm(Z—%) by continuation within ®*. These localised functions will be single-valued and will have a definite value for each value of the argument z corresponding to points within this region, though of course /,, fo, +... fm may have any number. We say that the g’s are a fundamental system of solutions of the equation, localised in ®, or briefly a fundamental system of solutions, it being understood throughout that such systems are so localised. From what has been said in §§ 2 and 3, g (2+ @)=a 9 (2), g (2+ o')=b g(2), where now we can speak of a and b as the matrices for the periods w, ’ respectively, since, as we have remarked, the g’s are single-valued. * We suppose z, a point within &. 390 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR Now consider any linear combination of the g’s, say F (2) =q gi (2) + Ae Jo (Z) +.» + Am Im (Z); or as we can write symbolically F(z)=r 9 (2). The function F(z) is of course only defined in ®; but since the g’s are expressible as power series about any point of the region, #’ is so expressible, and so can, like the gs, be regarded as a branch of a monogenic function localised in ®. Obviously F (z+ )=na g(2), F(z+o')=NXb g (2). Let us enquire whether we can choose the numbers X in such a way that F(z) is a doubly-periodic function of the second species. If #(z) is of this character we shall have F(z+o)=0F (2), F(e+o')=0 F(z), where @ and @’ are certain constants. Recalling that the g’s are linearly independent, it is seen that we must have nu Sy NSN r= SB) RH" Bs | r=1 It follows then that if the »’s are not all zero—and this is the only case of any use— 6 and @’ are necessarily such that the system of determinants / Ay, — 6 (he (hry Saoadod Chm by —6 Dy» shochs Die. ll, |] oy GOs | Wa beset: (om bay te ae cic Den | | l| a, a et — 0. Weenccrce Asn bs, big eae Dain [iices Pathcaetsparitenea sep vota seg ecen= Near aeae Bia ser a cron aaa Wz mete eter cvcre ce etetehe eee eter cater sale ate elaveierahe te ehetalata tors er sve oeeia e Riate etcletaletedess teisievelatelala chofeietelelofe sim taleleieeteemintcls state eter tam | yy a, a; Opy 0 Arm bya bi Dp». Sa a. b, m | a cacetnielesaiere(eleicie e einipie\sisicinjelsle ele vitis vleieini==/=jei> sicielelaiclo\si«.0/ojeinoaie\s/nlsin/sfels/o\elnle(e/e\e\els olsTo\ni@le\s\=/s)=/nI=jni“isisisici=isie eis || | mi Ging Chins OO O90 Amm a g Din Ome emicis (eis roe = q' or as we write for brevity || Gel) Bol Dany (el! \|; all vanish. Conversely if we can find values @ and @’ so that this actually does happen, we can choose at least one system of values X which are not all zero, and which satisfy the system of equations (i). We are led therefore to consider systems of determinants of this type. This is our object in the next few paragraphs, where, in view of future applications, we consider these DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 391 systems apart from the theory of differential equations, in order that we may appeal to it, not only immediately, but also at a later stage, when the circumstances are somewhat different. It would be interesting to consider the theory of these systems for its own sake; but as we have in view its applications to generalised Picard equations, we here confine ourselves to the bare minimum of properties necessary for our purpose. § 8. Let 2 and y* be any two matrices of m rows and columns, so that 2,s is the sth element of the rth row of 2, and y,, the corresponding element of y. The system of determinants lim, @ : Yim) & || will be called a conjoint system, provided that neither || nor | y|+ are zero and that a0) For example, the system | Qim> Gee Om a | considered in the preceding paragraph is conjoint. In fact we have seen that | a|+0, |6|+0 and as Chim be — Il, it follows that ab = ba. If for any values « of @ and «’ of @ the determinants | Zim, K > Yim ie | are all zero, we will say that «, x are an associated pair of roots of the system of equations ry OB die OF la@ It may happen in any particular case that not only all these determinants but also all their minors up to the (c—1)th inclusive vanish, whilst the vuth do not all do so; we then speak of “zc” as the index of the pair of roots (x, x’) of the above system. § 9. We proceed to prove that associated pairs of roots of conjoint systems exist and to give a method of finding them and their indices. Using the notation . | Am; 0 | for the determinant fy—O kp 73 Lim Boy Ts —\0) Dag) seen Gann Bay Tigh eoweseeas Lom ena LP pont encsae Imm — @ * The matrices a and J will be used to refer to period + |«| is a shorthand notation for the determinant matrices only. To prevent confusion weusexandy,and7 of the 2’s. so long as the theory of conjoint systems is considered + It is on account of this intimate connection between without reference to the differential equation. the x’s and y’s that the adjective conjoint has been used. Vou. XX. No. XVI 53 392 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR it is clear that if («, x’) is an associated pair of roots of \| tim, 9 = Jim || = 0, then « must be a root of \aaras 1 — Orne Suppose, therefore, that we take « to be one of the roots of this equation, and let the minors of order + in : | ! Zim, K be the first which do not all vanish; moreover, let one of these non-vanishing 7th minors be contained in the x,th, nth, ... m_,th columns and in the p,,,th, p,s.th, ... pmth rows of the determinant; 7, m, ... Mm; Pr;P2, +++ Pm» each consisting of the first m integers arranged in a certain order. Every set of values of the )’s which satisfy Nag = Dilip + Aa@ang bet AmnSmn, (Cr =1, 2) |}. mir) eee (ii) will necessarily satisfy the system Np = Ag@etk NaGerck cee Amime (= 15, 2) oa. 7) seen ec eeeanee (ili). Hence this latter system may be solved by taking any numbers we please for An eae es and then determining the remainder of the 2s as definite linear functions of them, say vee = kaXp, ae Kp, + woe te Mige Ap. (s=1, 2, ... m=) Now let us write Per = Mi Yar + AsYor + -+- + Am Ymr- Then if we take the system of equations (ili) and multiply them by %,, Yor «-» Ymr respectively and add, we obtain Prk = Dy (By Yar + Bye Yor + --- + Lim Ymr) + Ng (La Yar + Los Yor + --- + Lom Te) + 2X; ( ) + Am (Gm Yar + me Yor + +++ + mm Ymr)- Since oy = Ya, it follows that m Si = = yt Uts = =i) iM UYrt “ts- 1 Consequently Prk = (Yn Lig + Yao Loy + +++ + Yam Lmr) + Nz (Yor Bir + Yoo Voy + «+2 + Yom pee) + Am (Yma Ly + Ying Var Rigo SF Umm + 2mr ), or foro = fly Gap fee Dog Pe 5. Efe Demme ooo 222s voeen ee eee ncnsennnscenee (iv). * It may be unnecessary to point out that as | x,,,,0/=|c|+0 then «+0. Similarly «’+0. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 393 Now comparing the systems (iii) and (iv) we are not justified in assuming—as we might at first sight be tempted to do—that 2,/u, has the same value for r=1, 2,... m. All that we can conclude is that Mp5 eg = han Mp, + Kiso Bp, + 0 + Kesz bp, If however we have at [CES CIEE Ee Ror 5 (@) it clearly follows that Herts _ gy (ie een Oat) econpepconondcacencmerrrce (vi). Nori s § 10. Let us see whether we can choose the numbers 2» so as to ensure this. Take the + equations NY Np, + Des Yop, +..0.t+ Xp, (Yp,p, — 0’) +...t Press Ymp, = 0 (r = i 2, eiele: T) together with the system (ii); we have thus m equations; label them (vii). In order that we may be able to choose numbers 4,A,... Xm So as to satisfy (vil), we see that 6’ must satisfy wp, Yap, veeeveees Yip, Tin, Digs soon Gite. || 0......( Vill). | Yop, Yop Yop, Lan, Lop -sereeees 2m — | | oe Aaa aG xara nigunsivienicapasionecameca sees Bigg a hess cele vento cee aioneserisiele Ul | Yop (a) Yp,Dy sevesereteereee PyM, ceeeeeeeeeeceneaeeees Dp tm | | 4 Af 4 | Yrs, Yop, — OF -crrrserosnesennccersernceonrvertsssnacce d Lp, tim— | “ YP sewewe gene aanaemeannee weaeseneBsnae-maeecnereasseer nn MG ne Mines Ymp, Ornip. = 22-2 Ymp, — Tmn, c-eerereesraereeerees Soniieg = As the coefficient of (@’ in this equation does not vanish it is clear that it is of degree t in @. Let x’ be one of its roots, and suppose that «’ is written for @’ in the determinant which is the left-hand member of (viii), the oth minors are the first which do not all vanish. The system of equations (vii) may be satisfied by taking arbitrarily o of the 2's, say Xr A,,: and then determining the remainder as definite linear functions of these, say Xr Xo» Aen dor l,, Ng, + L,. Xa, +... +1, Na, (r=1, 2,...m—a); where q;,Q,---Gm are the integers 1, 2... m arranged in a certain order. Now whatever values have been chosen for Ag, Ag,,--- Ag the system of equations (iii) 1s satisfied since that labelled (ii) is; moreover, from the first set of the equations (vii) we have 394 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR From the remark made at the end of the preceding paragraph it follows that Ky / ie r=, 2... mm): ne ( ‘ and therefore that any system of values of the )’s, chosen as above, satisfies the system of 2m equations Ny = Aq Lip + AoW + Aggy + 0. + Am¥nr) ei apeton a aniteleear etek + AmYmr) We see from this that the system Weunn 228 Yims K’ || all vanish, i.e. with every root « of | Gig © | =O we can associate at least one nwmber x’, so that (x, «) are an associated pair of roots of the system of equations (Gan C3 Triey Fl =O The number «’ is of course a root of | Yam, O | = 0. It is not difficult to prove that the index « of the pair is equal to o. In the first place, since o of the 2’s can be chosen arbitrarily and then the rest determined as above, so as to satisfy the system of 2m equations (ix), we see that «>o. Again, since there is at least oth minor of a particular determinant of the system | Tim, K > Yims K I which does not vanish, viz. a oth minor of the determinant on the left-hand side of (viii) after @ has been replaced by «’, we cannot have u>o. The only remaining alternative is therefore «=o. § 11. One more point has to be attended to: we must convince ourselves that if we take all the roots of “ms d|=0 in succession and proceed as above, we determine all possible pairs of roots. Let (x, «’) be an associated pair of roots, no matter how determined, so that we have | Vimy K > Yims K | = 0. In the first place it is clear that « is a root of ee Penna Ale Vestn eer socoonocnadadde sucodosdadoAcanonon: (x). Suppose that the tth minors of | a», «| are the first which do not all vanish; and, adopting the notation of § 9, let one of these non-vanishing minors belong to the mth, nth, ... mn—th columns and the p,.,th, p,,.th, ... p,th rows. Since the determinant which the left-hand member of (viii) becomes after writing x’ for 6’, is one of the system l| Zim, # 2 Ym. X’ || it vanishes, ie. «’ is a root of (viii). The method of §§ 9 and 10 may therefore be regarded as a practical method of determining all associated pairs of roots. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 595 In passing we may notice that associated pairs may be arrived at in a similar way by starting with any root «’ of ptm lhl Ea ecbcacoe PREP PR RR EB ERE Stchcaacbone cdon sec stCeece (x1), and then determining « on lines analogous to the above; in fact the system is not altered if we interchange z’s and y's, @ and 6. The remarks we have just made prove con- clusively that the pairs arrived at in this way are the same as before. In particular, on referring back to the results of the preceding paragraph, it becomes apparent that if either of the equations (x) or (xi) has all its roots unequal, we shall have m distinct associated pairs of roots of the system Ween eS re ks (|| =a) The converse of this however is not true: for instance, if m=3 we have the three distinct pairs (x,, m,'), (x2, «), (x;, :), Which are associated roots of | «4-8 O 0 «4-8 0 Oa 03 | O t%—-@ O 0 Kk, —0’ 0 Ser eee ty i ee but «, is a double root of |m-6 0 One —0) O k-8@ 0 0 «,-—0 and «,’ is a double root of eta © O Ne jaa | 0) kK, —@ 0 | 0 0 ks — O | In any case however we are certain of the existence of at least one associated pair of roots. § 12. Having considered these associated pairs we pass on to consider a class of conjoint systems whose members are all closely connected with one another. Let p be a matrix of m rows and columns Pn Pir Pis --- Pim ; Pm Pm2 Pms +--+ Pmm which is such that |p|=R does not vanish. Suppose, moreover, that & and 7 are two matrices such that Ep = px, 7p = py. 596 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR The system When OS Gay Fl is said to be a transform of the system | Dims 0 : Yrm> 0 ||. Since ap = pi, é, | eal aE it is obvious that the latter is a transform of the former. We proceed to prove two theorems of considerable importance. §13. Theorem I, The transform of a conjoint system is a conjoint system. Adopting the notation already used, let || Zim, 9: Yim, 9 || || Em, 8 mm; 9 | its transform. In the first place we observe that be the original system, and [Ej=lz|, lyl=lal, so that |E|=0, |7|+0. Again we have np = py, and so NpL = pyk, le, since pe=E&p and wy= yx, nEp = pry. Consequently n&p = Epy = Enp. As |p|+0 we must have né = En, which proves the proposition. $14. Theorem II. The associated pairs of roots of any conjoint system are the same as those of all its transforms and the index of any pair is the same for each. Take a determinant of the original system which involves p columns of the a's and m-p columns of the y's, say | Bing Dypg srreeeeee np Yinp+1 Yinp+o +++ Yinm {§ | Von, Tang sereeeees ony Yonp4y Yonp4o 2+ Yorn Teeter eee eee eee eee ee eee ee eee ee ee eee ee ee rey ; | :adgagsenousnsauadadscoséocociene Urtanaty SOs agiceoneseceenenser Dee emma merce seer eee e eee eee eee eens sO Oe ese sesesessseresssssessessnsseceess | ®mny, Lmnry Pianyp Ymnpay cerrrsrseeeeeseeeeees Ynnn and call it D. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC Pu Pr Pis Pim Tiny Cony U3 “aaa Ul € Sosticnobacousasdoscenaaengono0 Tmny | Pa Bay os + Pom Bins (eS AAG Ete SON 0d SOOO EOdD SOC OOD CMOOCASOD ES Tro no — 6 Lmny Petareveletaiciercteiarsisiait'scieietclclesteiers We steteletntatale\sratetetef ota starela'a/aisvayate(alaia/sis{afaia\s\s)sselaje’s(eialalevsiatele):telelewlets eterislactatcisre\cisteisistiac ete | fa fesse Sodaneease fale OR ae Srancrcnocaaneene nce eeene Un eatapea ne O sevens esac Oe a i pti eeee ee | ee a ae RR ee | ee ek ee ee Pmi Pme -ersseees Pmm. | | Yat —-Yotig, 022s s essen sees sscsasseccceecccsscscrcesscsscessseveecs Yrm, m m ~ - = vs m S: , 4 = = = Piu®uny, — Ping 0 = Pru uns — Pine Castine Ea PiuYunp+1— Pinp +1 (le Wipgeee u= “= — m ™m m | . >> , y | > Peu%uny — Pen 0 = Pru uns— Peng GC) eceode = PouYunp+1— Penp +i ( caroao | u=l1 u=1 u=1 m m m >s 7 SS , SS hs = Prufun, — Prny e = Pru®uns — Prnre Oh Sasesc = PruYunpsi— Prap+ Oe sist | “= u=1 u=1 | x m m YP Ss . SS , i = Pmu®un, — Pmn, o> Pmu'Puns — Pmno Meksoke = PmuYunp+1— Pmnp +, (repbane u=1 u=1 u=1 Recalling that pe =p py=np, this determinant is equal to { m m m Ss a SS Ss f = Eu Pun, — Pin, 6 = Sepa — Ping OP rsauins = MuPunp+1— Pinp+1 CH ocpione u=1 u=1 u=1 m % m S & i = En Pun, — P2n 0 = fou Punz — Pens Ob waa © = AouPunp+1— P2enp +1 (aeanee u=1 u—1 u=1 m m m s ~) y ~ tS oof Dori — Prny g one Punsz — Pras Os see = DruPunp+1 — Prap+1 (open uU= u=1 u=1 ™m ™ m Ss: SS! a > Eonfan, — Pmn () 3 Sime Phe — Pmnz O Faseas = AmnuPunp+) — Prsaoele! ete u=1 u=1 u=1 The determinant just written may obviously be expanded as a sum of multiples of determinants of the transform-system Tf then («, «’) are an associated pair of roots of the latter, it follows that, as where D’ is the determinant derived from D by writing « for @ and «’ for @. other determinant of the system | Em, 9: 1m, O |. p| +0, we have 1D) = (9), | Zim, K = Yims K | COEFFICIENTS. 397 Similarly every 398 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR vanishes: in other words («, «’) are an associated pair of roots of the system of equations || tim, 9: Jim, O || =0. Now from a remark made above (§ 12) the system which is the left-hand member of the equation just written is a transform of || Em; 8: tim, % || and so by an application of the above reasoning we prove that every associated pair of roots of the former are a pair for the latter. We conclude therefore that the system of associated pairs of roots is the same for each. $15. We have next to prove that the index of the pair (x, «’) is the same in each. Take any nth minor of the product (RD). When the latter is written in the second form of § 14 it is clear that we can expand it as a linear function of the nth minors of determinants of the system | Sms () 3 Nims 0 |. Let » be less than v’ the index of the pair (x, «) for the system just written. Since the nth minors of determinants of the system | Eim, © > Mm; K | all vanish, it is clear that the nth minors of the product RD’ all vanish when it is written in the first form. Now the nth minors of a determinant of m rows and columns are «? in number, where suppose them to be d,, Ry, Py @y=12... ~) when formed from D’, R, RD’ respectively. We have : Po = Big Ras Aca es ty a Age As the numbers P,, are by hypothesis all zero it follows that OS RA RG AS eee eae 0= RA, + Rn Ay +... + RyA 0] Ryn A) + ReApt...+ hy, A m—l! . The determinant of the coefficients of the A’s is equal to R’+, where v= Ae m—n—1!n! and consequently does not vanish. It must therefore be that AMO peAgaeee el are all zero. In a similar way we prove that the nth minors of every other determinant of the system ne | Lim, K+ Yims K | vanishes. This proves that the index (c) of the pair («, x’) for the original system is >V/. Since the original system is a transform of the second it follows that te2h We have therefore «=, which completes the proof of the theorem. * Scott and Matthews, Theory of Determinants (2nd edition, 1904), p. 58. + Ibid. pp. 65, 66. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 399 § 16. The relevance of the work of the preceding paragraphs (§§ 8S—15) will be best appreciated as we proceed; but before doing so it will be well to make one or two observations which will be useful to us later and which at the same time bear testimony to its close connection with the subject before us. In §7 we were led to consider the conjoint system | Sa? 0 | where « and } are the matrices for the periods » and @’ respectively. We will call it the fundamental period system and the system of equations || im, 9: Din, 8 || =O we will call the fundamental system of period equations corresponding to the system of solutions g(2), g2(Z), --- Im(Z)- In § 11 we saw that this system of period equations has at least one pair of associated roots say («, x); and so, from what was said in § 7, we can choose at least one system of values for 2,, As, --- Nm (which are not all zero) such that 1 (s=1,2,... m) It follows that F(z)(=2 g(z)) has the property F(z+o)=x F(z), F(z+o)=«' F(z). In other words we have Picard’s theorem or rather a slightly more general one, viz, every generalised Picard equation has at least one solution, a branch of which considered as localised in the periodic region of the equation is a doubly-periodic function of the second species. §17. The following theorem throws light on the bearing which the theory of transforms has on our work. Theorem. Of any two fundamental period systems corresponding to two different fun- damental systems of solutions of the equation of § 1, considered as localised in the same periodic region ®, each is the transform of the other. Let one fundamental system of solutions be 92), J2(Z); --» Jm(2Z); and let any other be (2), y2(2); “se "Ym(2). The period matrices of the first system are a and 5: suppose that those of the second are a and 8, so that y(z+o)=a27(2), y(2+@')=B y(2). The g’s and ys being each fundamental systems we must have 7(2)=pg9(2), Vor 22 No: SVE 54 400 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR where p is a matrix such that |p| does not vanish. In less condensed form, the equation just written expresses, of course, that Yr(Z) = Pr Gi(Z) + Pre Jo(Z) + +++ + Prm Jn (2). (r =1, 2,... m) We have y(z+o)=ay(2)=24p9 (2), and again (2 + @) =pg (z+) = pag (2). Remembering that the g’s are a fundamental system, it follows that ap = pi. Mutatis mutandis we prove that Bp = pb. The theorem is thus established. Referring back to Theorem II, §§ 14 and 15, we conclude that all associated pairs of roots of a fundamental system of period equations and their indices are independent of the particular fundamental system of solutions from which they are derived—they are functions merely of the coefficients of the differential equation and the region ®. §18. We now take up the thread of our work as it was left in §7. Suppose that («, «’) are a pair of associated roots of the fundamental system of period equations Waar Oi Orga i105 and that o, is their index. We will call o, the first invariant of the pair, when considered in connection with the differential equation: ‘invariant’ implying the property at which we arrived in the previous paragraph, viz. that o, is the same for all fundamental systems localised in ®: the reason for using the word ‘first’ will become apparent as our method develops. From what has been said above (§§ 9,10) we can solve the system of 2m equations NGO ekcl= Nis Op tortie FN an rw’) Ai Gigak AgDas cfeceer deta Dis = ASIC by taking arbitrary values for o, of the X’s and then determining the remainder as definite linear functions of these. Without loss of generality we can suppose the o, 2s to be LissiNay escuee the other 2’s will then be determined by (Ged e857) enaace (x11), Nats = Keg hy Kee Poche No - (s=1, 2; :..m—G) Then taking any values we please for \4, Ay... o1) F(z) =A gi(Z) + Av Yo(Z) + «+. + Am Jm(2) Se (gle) + 5 hu gre) $= t=1 = Ae Ge(z) say; t=1 m-o; where Gt (Zz) =92(2) + = Ket Js+c, (2). s=1 * It is clear that the interchange of any two membersof corresponding rows of the system of period determinants. a fundamental system of solutions merely interchanges two a os DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 401 Since the equations (xii) above are satisfied we have F(z+o)=« F(z), F(z+o')=«’' F(z), whatever A,, A»...A,, may be. We must have therefore Gi (z+o@)=« G(2z) Gi(z+o')=«' G,(z). (ESI, 2 cco Ga) From the form of the functions @ it is clear that any non-evanescent linear relation among them would involve one of a similar nature between the g’s. As the latter are a fundamental system no such relation can exist among them. It follows that the @’s are linearly independent. We conclude that corresponding to any associated pair of roots (x, x’) of the fundamental system of period equations we have o, solutions of the equation which are such that one branch of each considered as localised in ® is a doubly-periodic function of the second species with multipliers («,«). Moreover these branches are all linearly independent. In particular if the equation is one of ordinary Picard type these solutions will be single-valued doubly-periodic functions of the second species, whose domain* is the whole plane with the exception of some or all of the singularities of the coefficients. The total number of integrals of this kind will evidently be equal to the sum of the first invariants of all associated pairs. It will be proved below (§§ 29, 30) that these are all linearly independent. Consequently whenever the sum of the first invariants is equal to m every integral will be a sum of multiples of doubly-periodic functions of the second species. For instance this will be the case whenever either of the equations | hee GSS || os COO has all its roots distinct (§ 11). §19. Take the system of solutions G, (2), G.(z), Sa G,(2), Jon (2), Go+2 (2), TO Jin (2). As any linear relation between these would involve one between the g’s it is clear that this system is fundamental. It will be convenient to say that any such system is normal with respect to the first invariant of the associated pair («, x’). Re-label this system PHAR UACA) eae lA) (eae) eae L(G) respectively. As before we shall have h(2+o)=ch(2), h(z+o’)=dh(z), where c and d are new period matrices. But now we have c,,=d,,;=0 provided r+s and T <6,, whilst cy = Cy =...... See ye 2 Oh 0 Spancce =d,,,,=« so that the new fundamental period system is * The domain of a single-yalued function is the com- other of the circles of convergence of its elements. pletely open region whose points are internal to one or 54—2 xk—O@ 0 Ol 25. See Ae eee 0 c«—O 0 One ee 0 ma) PM (i | OAR PPAR SS Sodan ona 0 0) MSO OPAscreer art ct oe sastseneere 0 0 0 eaanee 1 =O Nee eter dass 0 0 0 Oe Oe tones 0 Ul ) Ca,ti,1 Co,41,2 Co,ti,3 +++ Co,41, 0,41 — Bes: Co,4+i,m deta daaas occcccccecos da.+3, olen 6 oo dg 45.98 Geary De tentt enohcoonoacceSocosagendo2asosouGe Gh ism (Corre >a peia ies. esseoseneceereec erat aere de ta\en | ceecse sectretaenesteenee none meraeoeshtesuencterhes h-tcecgonernate-ceeerone et WiGeer (DORE To ERS ONnepencceecee: -qebeee Cnm—O Gana [fe PE Pret Nan ASaac oS bancc dmm—@ From the theorems proved in $17 and § 13 or by a direct appeal to the theory of § 5 it is clear that this is a conjoint system. But before we consider it in detail it will be necessary to make a further excursion into the general theory of such systems. § 20. Consider a conjoint system | fin (te Yim> ce l|, where 2,s=Y;s;=0 provided r+s and @g : Ynviym> co ||. We proceed to prove certain theorems about these components. * It is of course quite clear that (xx’) are an associated so that when we speak of a component it will be under- pair of roots of the system. stood that it is obtained from a conjoint system as above. + We use the word in no other sense than the present DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 403 § 21. Theorem I. The component of a conjoint system is a conjoint system. Adopting the above notation we have to prove that : a | Lati,ms Or: Yn, ms oh | is conjoint. Since wy = yw we have the relations m m Yr > Tuyus = > Yrufuss ( = saa HO u=1 u=1 which in the case before us become effectively m—-n7n m—n Sp — , = Un+rntuYntus= = Yntrntu ln+u,s- “u=1 u=1 (GSN yoo ty Sa PA Gan 1a) In particular we have = Shy son WH -n) m—n m—n Spr aS: ” =a N+, N+U Yn+u, 048 =. Yntr, n+u& N+U,N+8> (: s u=1 u=1 which proves the theorem. § 22. Theorem II. Any associated pair of roots of the component of a conjoint system whose first » rows are regular with respect to an associated pair («, «’) are an associated pair for the complete system. If («, x’) are the associated pair of the component the theorem is obvious: we confine ourselves therefore to the case in which this is not so. It will then follow that if (kk’) are the pair of roots of the component we may have k=« or k’=«’, but not both of these equalities can hold. Suppose that we have / + «, and let l| 21m>9 = Yim, 9 || be the complete system, Weeierale! 2 Omar | its component. We see at once that as k is a root of Frat ont) |=; it is also a root of Gia 7] =O Take any pth minor of the determinant | Pins k |. A little consideration will shew that if it does not vanish identically it must be equal to (x —k) x a(p+t—n)th minor of | @r44,m, k |, where tt. As, however, one particular tth minor of |am,k| is not zero, viz. one of the 7th minors of |@jii,/| which does not vanish, bordered by elements of the first n rows and columns of | am, |, it is clear that 7’ =r. Let us refer back to §§9, 10, and, in the notation there employed, suppose that the non-vanishing minor just mentioned is common to the nth, nth, ...m,-,th* columns, and * Of course 7 of these numbers are 1, 2, 3, ... 7. 404 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR the p,yth, p, th, ... pmth* rows. Clearly the determinant which is the left-hand member of (viii) § 10 is equal to (k—«)” x a determinant of the system || @sim)% : Ynsi,m) 9’ ||. As k’ substituted for 6’ in the latter makes it vanish, we see that (k, k’) are an associated pair of roots of [teal 2 Ope |=. The theorem is therefore proved. § 23. Theorem III. If of two conjoint systems one is the transform of the other, and at the same time the first n rows of each are regular with respect to the associated pair (xx’), then the component of the transform is a transform of the component of the original system. Suppose that aims O 2 Sime Il is the original} system and that Wisin) Sheng 7 | is its transform. Then by definition Ep = pz, np = py, where p is a matrix of m rows and columns whose determinant | p| does not vanish (§ 12). The first of these relations written at full length gives m m PS f= a faaneny (Chea P* G50 10)) ob sone qsoncono- (xi). 1 1 u= u= Hence if 7 ee hoyroo+s (2). Then whatever No,41, --- Ac,40. May be F (z+) =x’ F(z)+linear functions of h, (2), ... he, (2), P2+o)=«' F(z) + » » » It follows easily that we must have Ag+: (2 ae @) =K Ho, ++ (Zz) + Feiss (2), Hot (2 +0’) = « Host (2) + H’o,+1 (2); (t=1,2,... a2) where Ho,.,(2) and H’z,,,(z) are linear functions of h, (z), hy (2) ... he, (2). A little consideration will shew that we cannot have both these lmear combinations evanescent: for if any pair were, it is clear that we would have at least one more integral of the type of h,(z),h.(z),...h,(Z), which is independent of them. We would therefore be able to find a fundamental system such that the corresponding period system had its first o,+1 rows regular with respect to (x, «’). The index of any such system is clearly > o, which is impossible as we have proved that it must be o, for all fundamental systems (§ 18). For a similar reason it is easily proved that there exists no lnear combination of the functions Hz,,,(z) such that it and the same linear combination of the functions H’z,,;(z) both vanish. The functions H(z) are of course only defined in ®: but as we remarked in § 7, any such localised function may be regarded as a branch of a certain monogenic function localised in ®, § 25. From the form of the functions H(z) at which we arrived in the preceding paragraph, it is easily seen (cf. §§ 18, 19) that they are linearly independent, and further that the system of solutions h, (2), hs (2), ... ho, (2), Hoyar(2), Ho,12 (2), »-. Ho,+05(2), jenna. soo (3) is a fundamental system. ee DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 407 The members of this system may be re-labelled k,(2), ke(2), ..- ko,(2), ko,41(2); bo,42 (2), -.. ho,40,(2), «.- km(2) respectively. We will say that any system of this character is normal with respect to the Jirst two invariants of the associated pair (x, x’). As in the preceding case we shall have k(z+o) =ek(z), k(z+o')=fk (2), where the new period matrices e and f are such that (1) e¢s=frs=0 provided r+s and r 0, provided r = s and r Se, whilst C6341, 641 = Co, 49, 0,49 = +++ = Cai 069 = K, Sono = fo,+», O42 S++ = fo+0. = (6 If we consider the first derived period system of l| ms 9 : Sim, a |, viz. the system ll eo,41,m,9 : foy+r,m, 8" ||, it is clear that its first ¢, rows are regular with respect to the associated pair (x, x’). We use the notation | | €o3+0541,ms 0 : Fostoqss, my» a for its irregular component. Considered in connection with the differential equation we will call the latter a second derived period system with regard to the associated pair (x, «’). It is a very easy matter to see that any such second derived period s conjoint. The proof follows in fact from a double application of theorem I, § 21. For the system under consideration is the component of a first derived system: and we saw above by an application of the theorem mentioned that the latter was a conjoint system. Consequently a second application of the same theorem proves what we desire. In a similar way by using theorem III, § 23, we prove that all second derived period systems with regard to («, «) are transforms of one another. It follows there- fore by theorem II, §§ 14, 15, that the associated pairs and their indices are the same for all. § 26. It may or may not be that («, «’) are an associated pair of roots of the second derived system of equations . . | @o,+0.41, my @: Joy+0,41, m> || =0. If not then our process terminates: but if they are, let o; be their index. We call o; the third invariant of the associated pair (x, «'). After what has been said in the previous paragraph it will be sufficient if we merely remark that o, is the same for all fundamental systems which are normal with respect to the first two invariants of the associated pair (x, x’). Vou. XX. No, XVI. or or 408 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR From the general theory of §§ 9, 10, it appears that we can satisfy the system of 2(m—o,—,) equations Noytont1 Co,+oyH1, o,¢o,¢7 + No, too+2 Ca, +os+2, oytoztr t +: + Ne,to,+r (€c,+0,+7, O,+0,+7 Kk) a> 000 st An€m » O%tootr = 0, (r=1, 2,... m—o, — Ga) No,toy+1 rceacts » +0 +7 sF No, +o,+r Jo,+0,+2, o,+0,+7 ste teiete . , ate Ne, +or¢r (fo,+e,+r, ato,tr— & ) +...+Am M, +047 = 0, by taking arbitrarily o, of the X’s—which may be assumed to be Agiic.ii1, Aetertes ++ Ao,+o,+e,—and determining the rest as definite linear functions of these say No, poytogtr == | My, | Noy tort + | My, | Noyto,+2 Stale (ehots Mr, No,+-02+05" (r=1, 2, ... m—o, —o2.—@a5) Take the function F(z)= Ae, +0,414 9,464 (2z)+ Aetertrho torts (z) +... +Amkm (2); where the X’s are selected as above. Then os = F(z)= = No toptt Ac, +a,+t (2); where A , M—7,—F2-7s5 K oto,+t (2) = Ko, toytt (2) ate > Mest eooctocts (2). s=1 Proceeding in the manner of § 24 we arrive at o, solutions which are linearly inde- pendent and which have the properties represented by Ltrs (z +a) =k a: (z) ote Ree se (2), Ko,40,4t(2 + 0) = K' Ko, 40,40 (2) + Peron (2), where Kotor (z) and Ree tose (z) are linear functions of &, (z), kh. (2), ... ke,+0, (2). By reasoning analogous to that used in the corresponding case in § 24 we may shew that one but not both of these functions may be evanescent, or a linear function merely of k,(z), ko(z), ...e,(z): that is to say one at least must involve members of the second set of integrals (of § 24). In a similar way we shew that corresponding linear functions of the A’s and Ks have the same property. It is moreover easily shewn that the solutions Ki(2), bal2); 2 Mase (2), Kocora(2),, Kosara(2)y Kone a(@)y Mere eee constitute a fundamental system. § 27. The general course of procedure is now obvious: but before stating the complete result it will be useful to give a definition. We have seen that corresponding to an associated pair of roots («, «’) we have a number of invariants oj, o2, o;, &e. Let o, be the last of these so that («, «’) are not an associated pair of the sth derived period system with regard to (x, x’). The sum o,t+o.+...+0; is defined to be the multiplicity of the pair («, x’). From what has DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 409 been said above the multiplicity of the pair of roots is independent of the fundamental system from which we start. Slightly altering our previous notation our final result then is this:— Corresponding to an associated pair of roots (x, «’) of a system of period equations whose multiplicity is » we have a group of n integrals of the differential equation which are linearly independent. This group may be divided into s sets where s is the number of invariants of the pair. The first set consists of integrals w,(2), Wy. (z), .+. Wio,(Z), equal in number to the first invariant (¢,), which have the properties expressed by Wy (2+ @) = KW, (Z), Wy (2+ o’) = Kw, (2), (GPS, PA, say Gy) or as we may write symbolically W, (2+ @)=KU, (2), Ww, (2 +0’) =K'w, (2). The second set consists of the o, integrals w(z), we (z), ... Wog,(Z) Where o, is the second invariant of the pair. These have the properties Wey (2 + ©) = KWo, (Z) + Uy (Z) 2 Wor (Z + ©) = K'Woy (2) + Vy (2), (P=, Foc GA) Vr (Z), v4, (Z) being linear functions of the w,,(z)’s, both of which cannot be evanescent. Symbolically we may write Wy (Z +) = KW (Z)+ pnw; (Z) : W(Z+ o')=«K'w, (z)+ Pa (2), where p, p’ are matrices of o, rows and o, columns which are such that if we take corresponding rows of each, the elements of both cannot all vanish. The third set consists of integrals wy, (2), W(Z),...Ws0,(z2) whose number is equal to the third invariant of the pair. These have the properties expressed by Wor (2 + @) = K Wey (Z) + Vor (Z) + Uar (2) 2 Wor (2 +.) = ' Wey (Z) + V'y (Z) + Uy (2), Vo, (z) and v',,(z) being linear functions of integrals belonging to the second set of which one but not both may be evanescent: %,(z), wi,(z) are linear functions of integrals belonging to the first set. In the language of matrices W3(Z+@) = KWs(2Z)+ PsoWo(Z)+ ps, (2), Ws (Z + w’) = K Ws (Z) + p'soWs (Z) +p’ (2), where Prsy P'rs* are matrices of o, rows and o, columns: and so on. The properties of the rth set have for their expression W,(Z+@) = KW,(2)+ Pr, raWra(Z) + Pr, r2Wr—2 (Z) +... + Pri; (2), Wy (2+ @') = «Wy (Z) + Pr, ra Wr—-a (2) + p's, r—2Wr—a (Z) +...» + p'nW; (2), where p;,71W,4(2), p'r,7,2Wra(Z) are such that no two corresponding functions are evanescent and further no linear combination of either of the sets vanishes simultaneously with the same linear combination of the other (cf, §§ 24, 26). * The matrices p,,, p’,; are not independent (see § 34 below). 55—2 410 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR § 28. The functions at which we arrived in the preceding paragraph, it will be re- membered, are only defined in ®; but from the remark made in § 7 we see that they are branches—principal branches it wil! be convenient to call them—of n monogenic functions localised in ®. In general the remaining branches of any one of these functions when regarded as localised in ® will have properties different from those of the principal branch; but in the case of ordinary Picard equations the solutions are all uniform and so only principal branches arise. The group of localised functions arrived at in § 27 will be said to be normal for the associated pair (x, x’). Since every pair of associated roots can be treated in the same way there will obviously be a set of normal groups whose total number is that of the distinct pairs of associated roots of the fundamental system of period equations. ? § 29. The choice of the word “multiplicity” in § 27 will probably have prepared the reader for the theorem to the proof of which we now proceed, viz. that the sum of the multiplicities of the various associated pairs is equal to m, the order of the differential equation. First of all we shall require the following Theorem. No linear relation can exist between the members of the various normal groups of integrals. We have seen already that no such relation can exist within a group, so our attention may be confined to the case in which the members of more than one group are involved. Suppose then, if possible, that a linear relation, embracing the members of two.or more groups, exists, say Ware) a W (2) a eae eS) — 0 on gceneinr oan eeeeee (xvi), where W,(z), W.(2), ... Wi (2) are each linear functions of members of a group corre- sponding to a definite associated pair. It will be convenient to define the order of any such function to be p, where integrals of the pth set are the latest involved: eg. if the integrals involved are all doubly-periodic of the second species, the order will be unity. The relation (xvi) above written will be said to be of the pth order when p is the greatest of the orders of the various functions W,,(2). In the first place let us prove that every non-evanescent linear relation (xvi) of order p involves one of order p—1 provided that p> 1. From (xvi) we have W,(z+o) + Wi (e+e) +...+ Wi (e+) Sit Wi(z+o’)+ W.(e+o’)+...+ Wi(z+o)=0 Suppose that the order of W, is >1. An inspection of the results of § 27, shews that Wr (2+ @) = Kn Wh (2) + Vy (2), Wy (2 + @') = kn Wn (2) + Vn (2), where one at least of the functions V,,(z), Vn’ (z), which are linear functions of integrals belonging to the same group as the terms in W,, is of order less by unity (and no DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 411 more) than W,,(z). Suppose that V,(z) is such when W,,(z) is one of the functions of order p. From (xvii) we have K, Wy (2) + to We (2) +... + ee Wi (2) + Vi (2) +... + Vi (2) = 9, where one of the V’s at least is of order p—1. In a similar way since W, (2+ 2) + W,(2+ 20)+...+ W; (2+ 20) =0, we have 12 W, (2) + 2 Wa(Z) +... KEW (Z) + 24, Vi (2) + 2eo Vo (2) +... + 2e, Ve (z) + U2 =0, where U, is a sum of functions which may all vanish, but if they do not they are of order < p— 2. Generally we have Ky Wy (2) +... tee Wi(z) + 7K" V, (2) +... tree Vi (z) + U,=0...... (xvii), where U,. is of the same character as U). Now it may or may not be that all the numbers «, are different: if they are not we can group together those terms for which the multiplier is the same. Let this be done. Since the equations (xviii) are unaltered in form we can suppose them as written only now the «’s are all different, and the W’s and V’s may involve members of more than one group. § 30. Taking the system of ¢ equations obtained by writing r=1, 2, ... ¢ in (xvill) together with the equation (xvi) we can eliminate the W’s. We thus obtain a linear relation t A 2) es OO ainwcieieeareincss oyeiaela/se oteeraramowe stor (xix), =1 where U either vanishes or else is a sum of integrals of order 1) involves one of order p—1, and so by repeated applications whatever be the order of the relation (xvi) from which we start, we eventually arrive at a non-evanescent linear relation of order unity. § 31. Let the relation just mentioned be On Cao oO (a) eipag SOC (2) kU pongondbosonosnonsoccenescn aac (xx), where each of the ws is a linear combination of doubly-periodic functions of the second species which have the same multipliers. The multiplier corresponding to the period @ may be the same for more than one of the groups uw: if so, then we group together those of the w’s for which this happens. The relation (xx) then becomes V, (2) + Uo (Zz) +... +9; (z)=0, G<) where each member of the group v,(z) has the same multiplier for the period o. Since we have % (J+ 7ow) + (2+ 7rw)+...+0;(2+7e) = 0, it follows that Ky" 0, (Z) + Ko" Uo (Z) +... + Hj" 0; (y) = 0, for all values of 7: «, being the common multiplier for the period » of the terms of the group 2% (2). In particular we have the system of 7 equations (Zz) + U2 (Z) +...+ vy; (z) = 0, KV, (Z) + Kate (Z) +...+ 42; (Z) =9, K2U, (2) + KM (Z)+...+ KP Uj (2)=9, KV, (2) + KI? Vg (2) +... + Kj 0; (Z) = 0. The determinant of the coefficients of the v's is not zero* so we must have U, (2) = 0. (Ges PA dap) * Cf. § 30 above. DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 413 Now considering the left-hand member of any one of these relations we perceive that it is equal to W, (2) + We (Z) +... + We (2), where w,(Z), ...&;(z) are each linear combinations of integrals belonging to the first set of integrals of different groups. Since each of them had the same multiplier for the period » they must have different multipliers for the period w’. Applying an argument similar to that just used, it will follow that the w’s are all zero. But this is impossible, since as we have said they are linear combinations of integrals of the same group. Our initial assumption that there exists a relation of the form (xvi) is_ therefore proved false, Le. no linear relation can exist between the various members of normal groups. § 32. We are now in a position to prove the theorem mentioned in § 29. Theorem. The sum of the multiplicities of the various associated pairs of roots of a fundamental system of period equations is equal to the order (m) of the differential equation. Obviously the sum mentioned cannot be greater than m: for if it were the theorem of §§ 29—31 would shew that we had more than m linearly independent solutions of the equation which is known not to be possible. Suppose that it should happen that the sum is x where n «»+ Yeu, (2); on (2), 81 (2), a'ejale ela Sip, (2), Or (2), STOOL 8p, (2), SOT St, 74, (2), Xun (Z), Xr2(Z), «-++0 PCA Onl) onaebe Vor(Z); «= Xtyr,, (2) Wat (2), Vrn+2 (2), tee eeeee Vin (2), where the y’s are integrals of one group, for which there are ¢, sets, the first consisting of n, integrals, the second of m,, and so on, the total number of sets 4 being the number of invariants. Similarly for the 6's... y's. As for the w’s they are the m—n integrals mentioned above. Let the period matrices for this system be g and h, so that the period system is Ihe feapale/agal ane ||| Gacoonaeaqccconaccnasdesnortencescoradene (xxi). By blotting out the first » rows and the first n columns of g’s and h’s from this it is clear that we get a system of m—n rows and 2(m—n) columns; for brevity we call this the residuary system. It is easy to see by repeated applications of theorem I, § 21, that this residuary system is conjoint. It will therefore have at least one pair of associated roots (x, «’). There are two cases @ priori possible—either (x, «’) may be an associated pair of roots of the system (xxi), or they may not. The latter cannot be the case, for by repeated applications of theorem II, § 22, any associated pair of the residuary system is an associated pair for (xxi). Suppose then that («, «’) are an 414 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR associated pair both for the complete system and for the residuary. Without loss of generality we can suppose that the group of y's belongs to this pair. Repeated appli- cations of the theorem just mentioned shews that («, x’) are an associated pair for the system obtained by blotting out the first (m%+m)+...+7,) rows and the corresponding columns of the system (xxi), ie. a ¢,th derived system of (xxi) with respect to the associated pair (x, x’). By using theorem III, § 23, we prove that all such systems are transforms of one another: and further, from § 27 it appears that one of them has not (x, x’) for an associated pair of roots. It follows therefore from theorem II, §§ 14, 15, that (x, x’) cannot be an associated pair both for (xxi) and the residuary system. As we have seen already that there is no pair for the latter which is not one for the former, the only alternative is that the W's are non-existent, Le. m=m, which proves the theorem. Summing up our results thus far, we conclude that if we take normal groups corre- sponding to each of the various associated pairs of the fundamental system of period equations, they together constitute a fundamental system of integrals of the differential equation. § 33. The reader will now perceive the analogy that exists between the theory we have developed and that of the corresponding one dimensional problem in which the coefficients are simply periodic. It is true that our investigation is more cumbrous, but that is to be expected both from the nature of the case and from the fact that we have not had anything like the theory of equations and of elementary divisors on which to fall back. Moreover, we have not obtained anything analogous to Hamburger’s sub- groups. Two facts seem to conspire against this. The first is that the integrals are connected together in a much more intimate manner. For instance, if we consider the integral Wsy(z) of § 27, the functions v,(z) and v',,(z) may be distinct linear functions of integrals belonging to the second set, and similarly 2,,(z) and v',(z) may be distinct linear functions of integrals of the first set. Consequently, when we wish to express Ws, (2+ to + to’) as a function of integrals whose argument is z we will in general require two inde- pendent integrals of the second set and jour from the first, besides w;,(z). In the corresponding one dimensional case we would have to replace two and four by one and two respectively. The second fact is that the numbers oj, os, ... 7; may be an increasing set. All that we can be sure of is that Op SAO Ri ge SS In the first place we will shew that O2< 20}. Take the period system of § 19 and use the notation there employed. From the definition of co, we know that there is a certain o,th minor of a deter- minant of this system which does not vanish. Suppose that a of its columns have elements belonging to the first o, columns of the c’s and that 8 of them have elements belonging to the first o, columns of the d’s. The minor (J) in question may then be expanded by Laplace’s rule as a product of minors belonging to these 2+ 8 columns DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 415 and their complementary minors which belong to the remaining columns. These latter, if they do not vanish identically, are minors of determinants of the component system and cannot all vanish, since M does not do so. It follows therefore that as they consist of m—o,—a— £8 rows and columns we have m—o,-a—8s>t) which are the relations referred to. It is a very simple deduction from this to prove that if we take any product of matrices Pr,s; Ps; Psos, =+- Pept, (7>8>% >... >58)>1) and write dashes over any J of them, the product is the same as / ’ Pr, 3, P88, +++ P81 8p tay P 8p_a41 Sptae P 8110 8p—tig *** P apt Wor. Xexe Nos XeVie 56 416 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR § 35. Consider now the functions was, (2e-t@l, r@=32, fso-F4, where ¢ is the usual Weierstrassian € function and »={(o), 7 =€(o’). If we adopt the usual convention that the imaginary part of (=) is positive, we have u(z+o)=u(z), v(z+o@) =v(z)+1, u(zt+u')=u(z)+1, v(e@+o')=v(Z). Let us write AF =x F (2+ o) — F(z), A'F=*'3F(z+o)— F(z): then if ¢(z) is a doubly-periodic function of the second species whose multipliers are (x, x’), we have A (gu) =k 1g (2+) u(z+o) —$(Z)u(z)=0, A’ (du) = Kb (z+ 0’) u(z +o’) — (2) u(z)= $(2), and similarly A(¢v)=¢(z), A’ (dv) =0. More generally if we write _u(u—1)(u—2)...u—r+]1) u (r) r} ewe = 2) = 2 nel) aie r! ( where w= u(z), v=v(z), we easily prove that A (gu) =0, A’ (gu) = gu, A (gv) = gu"), A’ (pv) =0. § 36. Referring now to the results of § 27, and adopting the notation there used, it appears that the first set are doubly-periodic functions of the second species localised in ® and having multipliers («, «’). The second set satisfy 1 in A (wW») = re Pa Wy, (2), A (Ws) = = p 2 Wy (2). Consequently (§ 35) ( Wik, v A (us er Pat —— putts) =0, 4 AL, v \ A (w. Tae Pa; — = Parts ) =0, ie. the functions u v Wo (Z) — a PW; (Z) — 7 Putts (z) are doubly-periodic functions of the second species, localised in ® and having multipliers (x, x’), say they are W,(z). It appears therefore that the integrals of the second set can be represented by < Tes v W.(2)+ Paws (z)+ ig Pats (2), DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. or if we replace w, by W, in order to have a more symmetrical notation, we have aby, v W. (Zz) = Wa +o paW,(2)+ = pa W, (2), there being, of course, o, integrals of this kind. Again, for integrals of the third set, we have ie =e 1 Ws = = Poo Ws (z)+ 5 Pats (z) ea! Ww. ya WwW v Ww 2)) 1 Wits =P ( 2(Z) + pa 1(2) + pa 1(2 + Pa 1 (2) Se Wala) Spe Olt pen Ge pepe); == {Ps al (2 + Ps 1 (2)} + 7 PP 1 (2) +75 Po Pa 1 (2); = 5 , 1 , , U , , v ‘ similarly A’ (ws) = * (p's: Ws (z) + p’n Wi (2)} tie P's PW, (z) + ar P sz Par: From the first of these it is easy to see that v Ce & v Ui A (u,—2 paWe— 5 p'nWs—2 pa Wi— © p'aW, U (2) - U = ee P x2 Pin W, = v F . ye) z rea Ps2 P 2 W,- a Ps2 Pa W ) =(0, and similarly (since ps p'n = p's Px) A’ (w.- z Px»W.— =i =) K / We deduce from this that v Us, v hey % Ws (z) =W, (2) oF < Psa Wo (2) aoack so We (z)+ re pu W, (z)+ mi; p an Wy (z) U (2) ye) ee uv ae +7 Pe PaWi(2)+ "5 pm p'aWi(z) + Ps2 Px W, (2), K? where W;(z) are oc, localised doubly-periodic functions (multipliers x, «’). 417 § 37. Generally we say that we can form an expression for w,(z) by the following rule : Take any product Prs, Ps,8y P88, *** Psy _y,t> where PHS iS cos Se SU(DKEr =i) Write dashes over all but the first « of them. Then the symbolic expression for the integrals of the rth set is the sum of all possible terms y\P—® y@ (

where g(=p-—1) is the total number of p’s in the matrix product in the bracket (__), which corresponds to the general term above written, and y (= #—1) is the number of them undashed. =i It follows that A(P)=— at 2 Pr, Ws, Oe and therefore that A (p=) = 0 > se ventadedeee cose case oboe eee ee eee (xxii). Again, we see that ' 1 y P21) ym , : A (F) = re | sum of all possible terms like re Prs, +++? Cec p nea W, @ ’ where of course the terms corresponding to products of matrices which contain no undashed letters disappear (§ 35). In virtue of the remark made at the end of § 34 NV’ (F)= 5 {sum of all possible terms like y\P—2—2) y(z) , U ’ (x )P-2 KZ P rs, Ps, 8.°°* Psz8z15 P 8215 Snyn °°" P 8p at W, @} > where now there are p—a#—1 dashed letters at the tail of the matrix product. Collecting all terms for which p’,,, is the first term of this product, we see that u'r y'®) A’ (F)= af 71, Sl p" Ts) SS Ps, 35 *** Ps,8.45 Pini scaees = Pare iW, (9) ? where g(=p—1) is the number of p’s in the matrix product of the term in the bracket ( ) corresponding to the general term above written, and a is the number of them undashed, Hence a'F)=2 45 pt (Oh and therefore NA esl PYM) Soccich -Saoe AIS een Ios ae TO (xxiil). ————— EE DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 419 From (xxii) and (xxiii) it is clear that the functions w,(z)— F(z) are doubly-periodic functions of the second species, say W,(z). Then w, (Z) = F(z) + W,(2), which proves the formula to be true for w,(z). But we have seen that it is true for r=1, 2, 3; therefore it is true for r=4; and successively for r= 5, 6,.... The rule given above therefore suffices for the determination of the integral belonging to the various sets. ‘§ 38. We proceed to make an important deduction from the analytical forms at which we arrived in the preceding paragraph. It is embodied in the following : Theorem. The analytical expressions for the rth set of integrals belonging to the group associated with the pair (x«’) are all precisely of degree r—1 in w(z) and v(z). In order to prove this it will be necessary to shew that none of the functions ee wy el) a(%) 7 : lor (xe * (e Pr,r—1 Pr—a,r—2 +++ Pr—awti,r—a P r—x,r—a—-1 +++ P11 W, (2) are evanescent, i.e. that there exists at least one matrix , / Pr,r—a Pr-i,r—2 +++ Pr—w+i,r—a P r—a,r—a—-1 +++ Pa such that if g is an integer ,,1 1s such that the elements of its gth row are not all zero: denote this one (or either, if both have the property) by Pr,r1 SO that p,,,1 18 py,ya OF p’y,,1 as the case may be. An application of what has been said above shews that one at least of Pr, r—1 Pr—i,r—2> Py, r—1 Boa r—2 is such that the elements of its gth row do not all vanish; we can denote this by Pr,r— Pr-,r—2> where p;—1,r-2 18 either p,;,,-2 OY p'r+,r-2. Proceeding in this way we get a product Prjra Pr—1,7—2 Pr—2, 7r—3 +++ Pa» such that the elements of its qth row do not all vanish. This product is equal to / ’ Pr,r— Pra,r—2 +++ Pr—wyr—x P r—z-1 +++ Pas where « is the number of the p’s undashed (§ 34), and so the theorem is established. 420 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR § 39. Thus taking any generalised Picard equation whose solutions are uniform in a doubly-periodic region ® we can say, as a result of our investigation :— Corresponding to an associated pair of roots («, «’) (multiplicity m) of the fundamental system of period equations we can find m monogenic functions, whose principal branches are linearly independent polynomials in weal e—te}) ants (gh fre9-%). with coefficients which are doubly-periodic functions of the second species localised in ® and having multipliers («, «’). These polynomials divide into s sets where s is the number of invariants of the pair (x, «’), the rth of the sets consisting of polynomials (equal in number to the rth invariant of («, «’)) which are all of degree r—1 in w and »v. In the particular case of ordinary Picard equations, ® may be taken to be the whole plane with the exception of the singularities of the coefficients of the equation; and what has been said of principal branches applies to the complete function, the coefficients of the polynomials in w and v being now single-valued doubly-periodic functions of the second species. Still dealing with ordinary Picard equations, if we make the further limitation that its integrals are all regular at the various singularities of the coefficients, which are isolated so far as regards the finite part of the plane, then the coefficients of the polynomials in wu and v will be the meromorphic doubly-periodic functions of the second species usually considered in treatises on elliptic functions. 10 § 1. It will be observed that in the previous section, when we were considering the monogenic functions which were solutions of generalised Picard equations we had to confine ourselves to particular branches—principal branches as we called them. The remaining branches (possibly infinite in number) when regarded as localised in ® will certainly be expressible as linear functions of some or all of these principal branches, but that is about all we can say in general. Difficulties of this nature seem to shew that it is only by restricting the nature of the solutions that we will be able to attain definite knowledge of all branches of any given solution, and if this is true of a limited class of equations it will be @ fortiori true of the general class. The present section is devoted to the consideration of a class of solutions which, though very particular when looked at from a general point of view, includes the solutions of the ordinary Picard equation and of that of Halphen. As a preparation for this it will be well to resume for a little while the general considerations of (I, §§ 1—4). § 2. Suppose then that we take the equation of I, § 1, which is not necessarily assumed to be of the ordinary Picard type. Using the notation of the paragraph referred to, we saw that if we continued any power series P,(z— 2) to the point (z, + @) along a definite path (y) we were able to deduce from the resulting series one Q,(2— 2) which was a solution of the differential equation. The method we employed is obviously applicable if, instead of w, we take any DIFFERENTIAL EQUATIONS HAVING DOUBLY-PERIODIC COEFFICIENTS. 421 other period 2 (= nw + n’w’) where n and n’ are any integers positive or negative. It follows, therefore, that corresponding to each point z,-+2@ + n’w’ we obtain one or more power series* in (2—2,) which are solutions of the equation. From our general theory we know that not more than m of these are linearly independent; so there will be a certain minimum number ¢ of them such that all the others can be expressed as linear functions of these c. Any such ¢ series—say W,(z— 2%), W.(Z— 29), --. We(Z— %)—must obviously be linearly independent: we eall them a basic system at the point z, for the monogenic function f, whose initial element is P,(z—%). § 3. Suppose now that we take any other point ¢, which is not a singularity of f,. Instead of considering the elements of f, at the points z,+nw + n'a’ as above, let us study the elements of the function at the points €,+no+ no’. Let Z be a definite path from z, to &, which passes through no singularity of the coefficients, and let the series w,(z— 2%), w.(z— 2), ... We(2— 2) continued along this path by a standard chain become @,(z—£,), @,(2—&), ... @-(2—{): it is not difficult to see that the latter series are a basic system for the point ¢. In the first place, simce we can find periods QD), QO, ... A, such that w, (2 — (2) + 4)), we (2 — (4 + 2)), -.. WelZ — (4 +-)) are elements of f,, it is clear that o,(2—(0,+2,)), @.(z—(&+0.)), ... @.(z—(€& +)) are also elements of it. Again, let Il(z—&) be any element of f, whose base is {, and p be any path between & and +, © being any period nw +n’w’. When the element IH(z—{&) is continued along L from § to z we get a power series in (¢—2), say P(z—z,). Imagine that this last series is continued along the path ZLpL’ where JL’ is the path congruent with Z joining 2+ and +: the resulting power series P’ (z—(z+)) is equal to MW, (Z — (2) + Q)) + mews (2 — (2 + O)) +... + MeWe (2 — (4 +Q)), by hypothesis (§ 2) the m’s being certain constants, Now if IIl’(2—(&+)) is the power series, we obtain by continuing II (z—£,) from , to ¢,+ Q by a standard chain along p, P’(z—(z+)) when continued along L’ to (f+) will become II’(z —({,+ ©)); also the series w,(z—(2 +)) will become @, (z —(& + Q)) when continued along the same path. Hence Il’(z —(€ + Q)) = mo, (2 —(G& + O)) +... + me@.(2 — (& + Q)). * We shall describe these, in future, as the @ series for 2). 422 Mr MERCER, ON THE SOLUTIONS OF ORDINARY LINEAR If therefore y be the number for the point ¢, corresponding to ¢ for the point 2, we have y two new Q-series Wy, (Z— 2%) and w(z—~%). Then since W (Z— 2) = ky wy, (2 — 2) + keWp, (2 — 2) + «.-, = —— = we shall have W (Z— 2) =k, Wy, (2 — 2) + koWp, Wp, (Z — 2) +...- The left-hand side of this equality being a Q-series will be expressible as a linear function of the members of one transitive group. Hence if the group-letters at the points (Q+2,,+@) and (2+ 0,,+) — > be different, the series Wy, (Z— 2), Wp, (Z— 2) will be linear functions of members of two different groups, and so there must be at least one non-evanescent linear relation among the series a = Wp, (2 — 2), Whe (2 = Bo) sisises But such a relation is impossible, for it would involve the same relation between the series Wy, (Z—(2,+@)), Wp, (Z—(4+o)), «.. which by continuation along T° to 2, is easily seen to lead to the same relation between the series Wp, (2 —2o), Wp, (Z — 20), which are known to be linearly independent (§ 2). Our assumption that the group-letters at 2+Op,+0, %+2,,+0 are different is therefore false. We conclude that the group-letters to the right of the points corresponding to consecutive members of a chain, and hence of all members of a given group (say G,) are the same (say G,). Moreover, as the (Q-series, corresponding to every point marked with the letter G,, are linear functions of members of G, we see that everywhere we meet with the letter G,, that immediately on its right is G,. Mutatis mutandis 436 Mr MERCER, ON ORDINARY LINEAR DIFFERENTIAL EQUATIONS, etc. we prove corresponding results in regard to the group-letters to the left of G,, and also above and below it (in the same vertical line). § 21. There is another property of these group-letters which we shall require. It is that if a letter G, occurs at a particular corner of the net-work, then it must occur again in the horizontal and vertical lines through the point. For consider such a corner and the horizontal line through it. As we pass along this line in a particular direction, say to the right, we are bound to meet some letter (G,) twice; for there are only a finite number (7) of group-letters. Suppose that G, occurs at z,+no+n'o' and G, at the points HDtnotno, 2+nw+n'o. (n mm, G, must recur. In a similar way we prove that G, must be met again as we pass either to the left of, or above or below (2 +nwo + n'w’). § 22. Suppose now that the series w,(z—2), which is an element of f, (§ 17), is a member of the group G, which has say y constituents. The letter G, will of course be that marked at the point 2. Let z,+vw be the first point on the horizontal line through z and to the right of it at which the letter G, recurs. Similarly let 2+v'o’ be the first point in the vertical line through z, and above it at which the same letter is marked. 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