• 
 
 

 
 
 
 LIBRARY 
 
 OF THE 
 
 University of California. 
 
 GIFT OF 
 
 Class 
 
 
 
 
 
On the Curve y m —G[x)~O y and its Associated 
 
 Abelian Integrals. 
 
 DISSERTATION 
 
 PRESENTED TO THE BOARD OF UNIVERSITY STUDIES OF THE 
 
 JOHNS HOPKINS UNIVERSITY FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 BY 
 
 WILLIAM H. MALTBIE, 
 
 Baltimore. 
 
 1894. 
 
 • '"of rwt 
 
 PRESS OF THE FRIEDENWALD COMPANY, 
 BALTIMORE, M D. 
 

INDEX. 
 
 PAGE 
 
 Introduction 1 
 
 I. Bi-rational Transformation of the Curve y m — 6r (x) = 1 
 
 II. The Curve y m — B mn (x) = 0. Its Genus. Its Multiple Points 
 
 AND THEIR EQUIVALENT NUMBER OF DOUBLE POINTS AND CUSPS, 3 
 
 III. The most General Rational Function of x and y 8 
 
 IV. Integrals Connected with the Curve f=0. Their Reduc- 
 
 tion to Two Standard Types 10 
 
 V. Integrals of the First Kind. Their Number 13 
 
 VI. Periods of Integrals of the First Kind. Their Form, and 
 
 Reductions in their Number 18 
 
 Biographical Sketch 26 
 
 
INTRODUCTION. 
 
 The well-known work of Clebsch and Gordon on the Abelian Functions 
 assumes that the fundamental curve 
 
 f(xy) = 
 
 is such that only two of the values of y permute at each branch point, and that 
 the multiple points are either cusps or multiple points with distinct tangents. 
 There exists, however, a large class of curves included in the form 
 
 y ™—G(a?) = 
 
 (where G(x) is a rational function of x) which violate both these hypotheses. 
 They may, it is true, be made to satisfy these hypotheses by subjecting them to 
 a set of bi-rational transformations, but in the process they are deprived of all 
 their simplicity. 
 
 The purpose of this paper is to make a direct investigation of the integrals 
 associated with these curves and at the same time retain their characteristic 
 simplicity of form. 
 
 I desire here to express my gratitude and my sincere appreciation of the 
 great kindness and assistance of Professor Craig and of Professor Franklin, not 
 only in the preparation of this paper, but throughout my entire residence at 
 the Johns Hopkins University. 
 
 I. 
 
 Bi-rational Transformation of the Curve y m — G(x)=zO. 
 
 Consider the algebraic equation 
 
 *,-— G(aO = 0, 
 and write it in the form 
 
 * m -!t)= o > (1) 
 
 where B g (x) and B (x) are rational entire functions of x of degree q and r 
 
respectively. On this form Osgood * has made the following reduction. Put 
 
 y' «' 
 
 and (1) becomes 
 
 y ,m B r (aft) — t m + r -<iB q (a/Q = . 
 Put now 
 
 t=zax f + bt' V=l] 
 
 and we have 
 
 y>™B r (x') — (ax/ + b) "+'-«22 ff (a/)=: . (2) 
 
 Again, let 
 
 RrW 
 
 and we have 
 
 y" m -\B r (x')\'-'B m + r ^) = 0; 
 
 or, grouping together such factors of the product 
 
 as occur to the mth degree, 
 
 y" m — {B l (x')\ m B mir+1) . mi (*) = 0. 
 If finally we put 
 
 and drop the accents, we have 
 
 y m — B m{v + 1) (x)=0, (3) 
 
 where v is an integer >0. 
 
 This is Osgood's reduction. Examination of (2) will show, however, that 
 when 
 
 the function B m + r (a/) is no longer entire, and a different reduction must be 
 made for this form. The successive transformations 
 
 y»' ' -R q {x')' 
 give us the form 
 
 or as before, if 
 
 y m -B ml ,_ 1) (x) = 0, (4) 
 
 * Dissertation, Gottingen, 1890. 
 
where fi is an integer ]> 1. All possible curves of the form (1) are therefore 
 reduced by bi-rational transformation to the form 
 
 y m — B mn (x) = 0. »>0. (5) 
 
 The important things to note are : 1. That the degree of the polynomial 
 R is a multiple of the exponent of y, and 2. that not more than m — 1 of the 
 linear factors of R can coincide. 
 
 II. 
 
 The Cueve y m — R mn (x) = 0. Its Genus. Its Multiple Points and 
 their Equivalent Number op Double Points and Cusps. 
 
 We write the curve (5) in the form 
 
 y m — (x — a 1 )(x — a 2 ) . . . {x — a mn ) — 0, (6) 
 
 where the a's are real or imaginary constants not more than m — 1 of which 
 can be equal. These a's are all branch points of the function, and around 
 each of them all the m values of y permute in one or more cycles, according 
 as the number of coincident a's is or is not prime to m. Moreover, since the 
 a's are mn in number, it follows that the point infinity is not a branch point of 
 the function.* 
 
 Introducing homogeneous co-ordinates, we have 
 
 iz=mn 
 
 f=z mn - m y m —Y[(x — a i z) = 0; (7) 
 
 and from this form we see at once that no term is of degree, in x and z jointly, 
 less than m(n — 1); i. e., the curve has at x = z-=zO an m(n — l)-ple point, 
 whose tangents, as may easily be seen, are all coincident. 
 
 Again, the curve (7) has no other multiple points except such as arise from 
 the coincidence of two or more a's. For the necessary and sufficient conditions 
 that the point 0/3/2/ shall be a double point, and therefore the necessary con- 
 ditions that it be a multiple point of higher order, are 
 
 J-./VyV) = 0, 
 
 |/( !B 'y*')=o. 
 
 * Forsyth. Theory of Functions, p. 160. 
 
These three conditions are equivalent to the two following, 
 
 2 /m(n-l)-iym_.Q j 
 i = mn 
 
 ^(x — a 1 z)(x — a 2 z) .... (x — a i _ 1 z)(x-—a i ^iz) .... (x — a mn z) == , 
 
 i = l 
 
 which can be satisfied only by x'zziz' = (the multiple point at infinity), or 
 when, in connection with an identity of one or more other a's with a we have 
 x! = a^. 
 
 If then the a's are all distinct, the curve will have only the one multiple 
 point x = z z= , and its genus* will be 
 
 (mn — l)(mn — 2) ™ , QX 
 
 2>= v ^ ; — #, (8) 
 
 where E denotes the number of double points and cusps to which the 
 m(n — 1 )-ple point is equivalent. The coincidence of the tangents at this 
 multiple point makes the ordinary relation, "A &-ple point is equivalent to 
 J& (k — 1) double points," invalid ; and we must seek other means for the 
 evaluation of E. 
 
 Assume all the a's to be distinct. The function has then mn branch points 
 at each of which all the values of y permute in a single cycle, and these are the 
 only branch points. The genus of the curve is therefore 
 
 (m — 1) (mn) — 2m + 2 (m — 1) (mn — 2) , Q >. 
 
 P— 2 ~~ 2 ' { } 
 and we have 
 
 E= m(mn— 2)(n— 1) ^ ^ 
 
 When, however, some of the a's coincide, the resulting multiple points 
 have common tangents, the formula \h (k — 1) again fails, and the method 
 above employed gives us only the total number of double points and cusps to 
 which the two multiple points are equivalent. We must accordingly make a 
 direct investigation of these points, and determine whether the number of 
 double points and cusps to which they are equivalent is variable with the a's. 
 
 Making y = the line at infinity, and returning to the non-homogene- 
 ous form, the equation of the curve becomes 
 
 v m(n — 1) 
 
 — It (x — a ( z) = 0. (11) 
 
 *The mathematical faculty of the Johns Hopkins University have agreed to use the 
 term "genus" in place of "deficiency." 
 
The multiple point is now at the origin, and the line zz=0 is the common 
 tangent. For the evaluation of the multiple point we will follow the method 
 introduced by Cayley,* and form by Newton'sf method the expansion for each 
 branch in the neighborhood of the origin, 
 
 z=zAx a +Bx^ + , 
 
 where A, B, . . . . are constants and 
 
 i<«</3<r< — 
 
 To determine A and a we substitute in (11) 
 
 z z= Ax a , 
 which gives us the form, 
 
 Jm(n-l) x *m(n-l) _ x mn _|_ A Z (d^ X mn ~ l + a — A 2 Z (a^) &>*-* + **+. § § > _> Q . 
 
 The term x mn has its exponent independent of a , and none of the following 
 exponents can be made equal to it so long as we have «^>1. Therefore we 
 must have 
 
 am(n — 1) — mn, 
 n 
 n — 1 
 and 
 
 Ip-rri 
 
 A m *- l *=zl M A^e^^K p=zt,2,&, m(n — 1) . 
 
 The curve has evidently m(n — 1) branches corresponding tothem(n — 1) 
 values of A. For the farther development of any branch we assume 
 
 2p7ri n 
 
 z z=z e m ( n — 1 > x n — i + Bx$ , 
 and again substitute in (11). The result will be 
 
 X mn + m(n— 1) B&^^) X^l [m{n ~ l) ~ l] + fi +. . . . + B m{n - 1] X ^{n-\)_ x mn 
 
 + Z{a i )\J^-^) x mn - l +^r x -\-Bx mn - l ±P\ = 0. 
 
 Now the term 
 
 Z{a i )e^{n-\) x mn - 1 + ^r 1 
 
 has its exponent independent of ft and none of the following exponents can be 
 made equal to it so long as /9 ^> — — — ^> 1 ; and the least of the preceding expo- 
 
 * Collected Mathematical Papers, Vol. V, p. 520. 
 t As given by Salmon, Higher Plane Curves, p. 44. 
 
nen ts is — ^r [m(n — 1) —1] + /5. We have therefore 
 
 4pTTt 
 
 n^+J. - R __ —I{a i )e^^-i) 
 P — n —l' m(n— 1) 
 
 If, as can easily be shown, the terms of the series all have n — 1 as the 
 common denominator of their exponents, then each of the branches of the curve 
 is an (n — l)-tic branch; i. e., consists of n — 1 partial branches given by the 
 developments 
 
 2 (P) _-. e 2ni lm(n-l) + r^IJ x ^=l _ ^ ( a i) ^ [m (n - 1) + ,T=1 J ^T\ -}-...., 
 
 1 m(n 1) 
 
 .r p ■ 2 "I n y , \ f 2p . 4 "| n-fl 
 
 g (p) e L" 1 (n— 1) **" n— lJ^n — 1 ^ W e Lm(» — l) + n — lJ ^.n — 1 I 
 
 2 m(n — 1) * ' * * ' 
 
 n_1 ra(w — 1) 
 
 The ensemble of partial branches belonging to the m(n — 1) total branches 
 will be obtained by giving to p in the above system the values 1, 2, 3 ... . 
 m(n — 1) . If now we form all possible differences 
 
 Z 8 Z 8' t 
 
 and denote by M the sum of the exponents of their first terms, we may, following 
 Cayley, say that our m (n — l)-ple point is equivalent to }[if — 3m (n — l)(w — 2)] 
 double points and m (n — l)(w — 2) cusps. 
 
 The evaluation of M, while possible in any particular example, is entirely 
 too complicated to be attempted in the general case. The important thing to 
 notice is that since the expansions for the partial branches differ only in the 
 exponents of e , the exponents of the first terms of the differences z { 8 r) — 2#" } 
 will be independent of the relative values of the a's ; and therefore that the 
 number of the double points and cusps to which the m(?i — l)-ple point is 
 equivalent is unchanged when two or more of the a's coincide. If then this 
 number can be found when all the a's are distinct, it is found for all cases. 
 But this has already been done. We can therefore now affirm that the 
 
 m(n — l)-ple point is equivalent to — ± ^ ' ordinary double points 
 
 and cusps. 
 
If h (k < m) of the a's coincide, the method of page 4 gives us the com- 
 bined equivalence of the two multiple points; and the permanence of the 
 equivalence of the first enables us to find at once the equivalence of the second. 
 
 In particular, if B mn (x) has a factor of the form (x — a*)*, where k is 
 prime to m , we will denote its equivalence by JS^ . Then 
 
 P 
 
 (mn — l)(mn — 2) m(n — l) (mn — 2) ™ 
 
 But the function has now mn — k + 1 branch points, around each of which all 
 the values of y permute in a single cycle. Its genus is therefore 
 
 P 
 
 __(m — l)(mn — & + !) — 2 m + 2 
 __ _ 
 
 . „ (m -!)(&- 1 ) 
 
 M V 2 ' (12) 
 
 On the other hand, if k is not prime to m, let k=.lp , m=zXp , where I is 
 prime to X, Then the function will have mn — k branch points at which all 
 the values of y permute in a single cycle, and one where they permute in p 
 cycles. Its genus will therefore be 
 
 (m — l)(mn — k) + m — p — 2m + 2 
 and we have 
 
 *= 2 
 
 E, = (fn-m-l) + P -l > (13) 
 
 Moreover, a second Newton expansion will show that the equivalence of a { is 
 unaffected by the relative value of the remaining a's. We are therefore able to 
 find the equivalence of all the multiple points of 
 
 ym _ (J,. __ a ^k x ^ __ a2 )* 3 fa __ a ^fc a = } (14) 
 
 where 
 
 S? k { ss mn 
 
 i=l 
 
III. 
 
 The Most General Rational Function of x and y. 
 
 The most general rational function of x and y, when they are connected by 
 the relation (5), is of the form 
 
 A'tf*- 1 + B'y m - 2 + +I/y + M' 
 
 Ay" 1 ' 1 + By m ~ 2 + + Ly + M ' 
 
 (15) 
 
 where A', E, .... M', A, B, . . . . if, are arbitrary rational entire func- 
 tions of x. The first reduction to be made is to render the denominator a 
 function of x alone. 
 
 When m = 2 we have the hyperelliptic case. When m=3, Thomae* 
 makes the reduction by multiplying both numerator and denominator by 
 
 (Atfr* + Byv + 0)(Ayh + Byz* + C) , 
 
 where 
 
 2ni 
 
 „ 3 
 
 In dealing with the general case we may either extend this method of 
 Thomae's, or follow the method used in the general theory of Abelian Func- 
 tions. For this denote y m — R mn {%) by /and 
 
 Ay™- 1 + By m ~ 2 + + Ly + M by <p . 
 
 Then the product of <p by a factor which renders it rational in x alone is, as is 
 well known, 
 
 1 — R 0.. 
 
 10 — R 0. . 
 
 fy m ~ 3 
 
 ABODE M 00 0...;^ w_1 
 
 m — 2 
 
 A B C D L M 000 
 
 n 
 
 00 00 A B QBE 
 
 * Ueber eine specielle Klasse Abelscher Functionen. Halle, 1877. 
 
9 
 
 Multiply the first column by y m and add it to the (m -|~ l)th column. Multiply 
 the second by y m and add it to the (ra + 2)nd, etc. Then, since /=0 we 
 have for our rationalizing factor 
 
 J = 
 
 M 
 
 Ay m 
 
 By m . . . 
 
 . Jy™ 
 
 Ky m 
 
 ym-1 
 
 L 
 
 M 
 
 Ay m . . 
 
 . ijr 
 
 Jy m 
 
 ym — 2 
 
 K 
 
 L 
 
 M . . 
 
 . Hy m 
 
 ly m 
 
 ym — 3 
 
 D 
 
 E 
 
 M 
 
 Ay m 
 
 y' 
 
 B G 
 
 D 
 
 L 
 
 M 
 
 y 
 
 A B 
 
 
 
 K 
 
 L 
 
 l 
 
 a determinant of order m whose law of formation is evident. 
 
 The same result may be obtained by assuming a factor of the form 
 
 performing the multiplication, and equating to zero the coefficients of 
 
 Vi y\y* y m " 1 
 
 which will give us m — 1 homogeneous equations for the determination of the m 
 quantities «, /9, y, . . . . fi . 
 
 All of the methods indicated must give (except for a possible factor in x 
 alone) the same result. For given any two factors F(xy) and $(xy) which 
 will reduce <p (xy) to a function of x alone, we have at once 
 
 <p(xy) F{xy)z=(/; 1 {x) ) 
 <p{xy) 0(xy) = <ff 2 (x), 
 
 from which 
 
 w-gg#<w. 
 
 The most general rational function of x and y, when they are connected 
 by the relation / (xy) =. , will be of the form 
 
 where A l} A 2 , . . . . A m> and the product d(p are functions of # alone. 
 
10 
 
 IV. 
 
 Integrals Connected with the Curve /=z0. Their Reduction to 
 
 Two Standard Types. 
 
 All possible integrals connected with the curve /=. may now by the use 
 
 of the multiplier J %- be reduced to the form 
 dy 
 
 { A l3 r^+A^r^+....+A mdX9 (16) 
 
 J rf 
 
 dy 
 where A t and X are rational entire functions of x. 
 
 From this point we have two analogies to follow. The curve 
 y m — B mn (x)=z0 evidently occupies a middle ground between the hyperelliptic 
 curve y 2 — Ri(x)=zQ, on the one hand, and the general Abelian curve 
 F(xy)=z0 on the other. 
 
 We will follow first the analogy of the hyperelliptic integrals and reduce 
 the general integral (16) to integrals of two more simple types. 
 
 The theory of decomposition of rational fractions enables us at once to 
 reduce (16) to a sum of integrals of the two forms 
 
 f P * (%>**, and f-J2lM*!L 
 
 dy dy 
 
 where P x and cp x are rational entire functions. These in turn by simple separa- 
 tion of their terms and use of the relations f~0, and §£- = my™- 1 , are 
 reduced to the two forms 
 
 [ Q{x)y a dx m) f <p{x)y«dx nft . 
 
 J my™- 1 ' { } iix—aymy™- 1 ' {i * } 
 
 where a is a positive integer less than m. 
 
 Consider the form (17). Let the degree of Q(x) be denoted by q, and let 
 L (x) be a rational entire function of degree I. If then we subtract from 
 
 ( Q(x)y*dx 
 J my™- 1 
 
 the expression d (L (x) y a + 1 ), 
 
 we shall reduce the integral to an integrated part and to the new integral 
 
11 
 
 The relation /= gives us 
 
 dy _, R'{x) 
 dx my m ~ 1> 
 
 and by means of this and/=zO we can put the integral in the form 
 
 [ lQ(x)-mLi{x)R{x)-(a+\)L{x)R>(x)-\y« ^ 
 J my™— 1 
 
 If now we take I = q — mn + 1 > the last two terms in the bracket become a 
 polynomial in x of degree q. We can now so determine the q — mn -f- 2 
 arbitrary constants in this polynomial that the entire expression in the 
 brackets becomes a polynomial in x of degree q — q-\-mn — 2 = mw — 2 . If 
 now we separate the terms of this polynomial, we shall reduce all the integrals 
 of the form (17) to the form 
 
 [x^y a dx /3<mn — 2 ,-.qx 
 
 J my m ~ x ' a<mn — 1 * ' 
 
 and these we will call integrals of the first type. 
 
 The form (18) divides into two classes according as a is or is not a root of 
 R mn (x) = . When R mn a^O, we subtract from the integral (18) an expres- 
 sion of the form 
 
 where G is an undetermined constant. The form (18) then reduces to an inte- 
 grated part and to the integral 
 
 rr , ''■ C(a + l)y a (x — a)^fLdx — (J— l)Ch*+ l dx 
 
 [f <p(x)y a dx \ ~r jj \ J dx \ J y 
 
 J L{x — a) l my m - 1 {x — a) 1 
 
 __ tt<p{x)— C(a + 1) R'(x) (x — a) + (I— 1) CmR (a;)] y« dx 
 J (x — a) l my m ~ l 
 
 If now C be determined by the relation 
 
 <p (p) + (I — 1) CmR (a) = , 
 
 the above integral reduces to one of the form 
 
 [ 0(x)y a dx 
 }(x — a) l - 1 my m - i; 
 
 and this process may evidently be continued till l=z 1 , i. e. all the integrals 
 of the form (18) reduce, when R mn (a) ^p 0, to the form, 
 
 < F{ ifi-i - «^™-i- (2°) 
 
 (x — a) my m x ^ 
 
 These we shall call integrals of the second type. 
 
12 
 
 If, however, a is a root of B (x) = of order h , we subtract from the inte- 
 gral (18) an expression of the form 
 
 d ( ^ a+1 ) 
 
 ,(x — a) 
 C(a + IX*— a) y \Mdx-- 0(1 + k — 1) y +1 <& 
 
 ^ (« — a) l + k my m - 1 
 
 Put now 
 
 i2(a0 = (s — aYG x {x), 
 R>(x) = (x — af-^G^x), 
 
 and this fraction becomes 
 
 [g(g + 1) ^ (a?) - <?(* + ft - 1) gg, fefl ^ 
 
 We are therefore able to reduce the form (18) to an integrated part and to the 
 integral 
 
 La [yW-g^ + ilgiW + g + fe-il^W)] dx 
 
 ) y ' (x—aymy™- 1 
 
 Since G 1 (a) zj= , and Gr 2 (a) ;£ , we can so determine G by the relation 
 
 p (a) — 0{ («+ 1) G 2 {a) + (l + k — l)m0 1 (a)} = 
 that this integral shall reduce to the form 
 
 f F(x) y a dx t 
 ](x — a) l - l my m - i; 
 
 and the process may evidently be continued till we are reduced to the form 
 
 [ F{x)y a dx 
 J my™- 1 
 
 In this we may reduce the degree of F by the method already given and find 
 again only integrals of the first type. 
 
 All the integrals connected with the curve f=zO are now reduced to the 
 two types 
 
 I [rfjfdx /3<mw- 2 ] 
 
 IL f F(x)y*dx R(a) zfz 
 
 }(x — a)my m - 1 ' a <m — l 
 
13 
 
 y. 
 
 Integeals of the Fiest Kind. Theie Numbee. 
 Consider the type I and put it in the form 
 
 m — 1 ' 
 
 )m(R(x)) m 
 When x becomes very great this becomes comparable to 
 
 L^ + an-n(m-l)^ ? 
 
 which remains finite for x very great so long as /3 + an — n(m — 1)<C — 1« 
 This inequality for ft positive is possible for all values of a < m — 1. In order 
 therefore to find the number of integrals of the first type which remain finite 
 for x very great, we have the formula 
 
 a = m — 2 
 
 Vr n (m—l)(mn — 2) 
 / \mn — an — n — 1J = - -^ - • 
 
 a = * 
 
 The hyperelliptic analogy would lead us to call these l m )\ — "—J integrals, 
 
 "integrals of the first kind," but the analogy fails at this point. In the 
 hyperelliptic case m = 2, R t (x) is therefore reducible by bi-rational transfor- 
 mation to the form 
 
 f — R 2n (x) = > 
 
 where R 2n (x)hsiS no multiple factors. The curve has therefore no multiple 
 points except the 2(n — l)-ple point at x=zz=zO. At this the integrals 
 remain finite, and therefore the hyperelliptic integral which remains finite for x 
 very great remains finite throughout the entire plane. 
 
 In our case, if x — a is a multiple factor of R mn (x) , the expression 
 
 — \^_ x is, in general, infinite of an order \1 at the point xz=a; and the corre- 
 sponding integral is therefore not of the first kind. If, however, R mn (x) has 
 no repeated factors, we have * — ' ^ ~^—L integrals of the first kind, and 
 
 this is equal to the genus of the corresponding curve /=0 . 
 
 If a is a root of R mn (x)=zO of order k, we take the general integral of 
 the first type 
 
 ( Q(x)y a dx . (21) 
 
 J my" 1 ' 1 ' v ; 
 
14 
 
 and subtract from it 
 
 At 
 
 ,{x — a) 
 
 d ( (¥ +1 \ 
 
 C( a + l)y(x— a) d -ldx — 0(k— l)y« + 'dx 
 
 (XX 
 
 (x — af 
 
 _ y [(7(g + 1) (x — a) R'— Cm (k — 1) #] da; 
 ~" (a?— aYmtf*- 1 ' 
 
 which, if we define (xt and 6r 2 as before, is equal to 
 
 y" \_C(a + 1) & 2 (a?) — Cm {h — 1) G t (x) ] cfc 
 
 If now C be defined by the relation 
 
 Q (a) -<?[(« + 1) G, (a) + m (h- 1) fi^a)] = , 
 
 our integral reduces to an integrated part and to an integral of the form 
 
 M x — a) Qi(x)y a dx 
 J my m - 1 ' 
 
 where Q x (x) is of the degree mn — 3 . This reduction may evidently be 
 continued till we have the form 
 
 [ (x—af- 1 Q k _ l (x)y a dx 
 J my™- 1 
 
 Denote now by the symbol [2] , where z is a real number, integer or fractional, 
 the greatest integer contained in z. We note, first, that 
 
 *_i 5 [A (m _i_ a) ], 
 
 and therefore, second, that all our integrals of the first type reduce to a set of 
 integrals of the form 
 
 J m ym-l-a * 
 
 which are finite when x-=za . Separate this into its terms, put m — 1 — a=id , 
 and we have the integrals 
 
 Ux — a)^xfidx ^ mn ~ 2 ~ [£J 
 
 J m f 0<^ro — 1 
 
15 X^ 
 
 In order that these may remain finite for x very great we must have 
 
 The number of integers satisfying this condition is 
 
 =1 
 
 When k is prime to m a known theorem gives us 
 
 8 = m — 1 
 
 (h — l)(m—l) (22) 
 
 m 
 
 2 
 
 a quantity which we have already found as the equivalence of a &-ple point on 
 f= , when k is prime to m. 
 
 If k is not prime to m, put h = fy? and m = X[> and an immediate extension 
 of the above-mentioned theorem gives us 
 
 \ = m — 1 
 
 yp<P__(& — l)(m — l) + /o — 1 
 
 5 = 1"- 
 
 
 2- - , (23) 
 
 which is the equivalence of the &-ple point when k is not prime to m. 
 
 If similar reductions are made for all the multiple factors of R (x) , the 
 number of integrals which remain finite throughout the plane, i. e. the number 
 of integrals of the first kind, will evidently be 
 
 (^■2)(»-l) ,^i =J>| (24) 
 
 2 
 
 where %E t is the sum of the double points and cusps equivalent to those 
 multiple points of f=.0 which result from repeated factors of R(x), The 
 number of integrals of the first kind is therefore in all cases equal to the 
 genus of the curve. 
 
 We have now formed p integrals of the first kind, and they are evidently 
 linearly independent. But if we ask for the most general form of an integral 
 of the first kind, and whether there may or may not be more than p of them, 
 we must turn our attention to the more general theory of Abelian Functions 
 
16 
 
 and follow the analogy which it presents. We know that the most general 
 integral connected with the curve fz=. is of the form 
 
 f JMT-'-Mar >+■ -+A, _.*,. (25) 
 
 (x — a^i (x — a 2 ) ? a . . . . (x — a v ) l vmy r 
 
 What are the conditions under which this will always remain finite ? 
 
 Consider the point x=na i . The integral is evidently infinite at this point 
 unless the numerator vanish also. Two cases arise corresponding to the two 
 conditions R («J zfz , and R (a t ) ss . 
 
 When R {a t ) zfzO , y has for xzzL^m values all different and all different 
 from zero. In order then that the integral remain finite when x-=.a i it is 
 necessary and sufficient that the numerator, a polynomial in y of degree m — 1 , 
 vanish for m different values of y. Its coefficients must therefore all vanish, 
 i. e. the A's must all have a factor x — a t ) and the integral reduces to the form 
 
 Atf*- 1 + .... + 4 
 
 I (x — a^ ....(# — 0-i) li ~ l .... (a? — a„) l vmy- 
 
 m — 1 
 
 dx . 
 
 A repetition of the argument will evidently remove from the denominator all 
 the factors (x — a t ) , where R (a { ) :£ . 
 
 When a t is a root of R (x) = of order k , we write 
 
 R(x) = (x—a i ) k G(x) ) 
 and put for y its value 
 
 (R (x)) m =z(x — a t ) m ( G {x)) m . 
 
 The integral now becomes 
 
 k(m — 1) tn — 1 fc (m — 2 ) m — 2 
 f ^l(g — «,) » (G(x))m + A 2 (X—0. 1 ) rn (g fr)) m + 
 
 J (x — aj* (x— «,) i+ ^ - ; ....(a? — a,)V(0(aO) " 
 
 In order that this remain finite for x=za i it must reduce to the form 
 
 ( <p (x) dx 
 
 where <p (x) is finite for x-=za iJ and k is less than unity. The numerator must 
 
 therefore contain (x — a t Y L m J as a factor. For this it is necessary and 
 sufficient that we have 
 
17 
 
 A 1 =i(x—a i ) l iB l9 
 A 2 =(x — a t ) lmJ B 2 , 
 
 A m = (x-a i ) L - lB m . 
 
 (27) 
 
 The repetition of this process will remove from the denominator of (25) all the 
 factors (x — « f ), where R(ai)=:0; and (25) accordingly takes the form 
 
 U ll r-* + A4r- + .... + A m <lx . (28 ) 
 
 J my™' 1 v 
 
 This must still be examined for x very great and for those values of x 
 which satisfy my m ~ 1 = . For x very great we see at once that, if (28) is to 
 remain finite, we must have 
 
 A t =0, ~\ 
 
 A % of degree n — 2 
 J (29) 
 
 A m of degree (m — 1) n — 2 . J 
 
 The points which satisfy my m ~ 1 =z are the roots of R (x) z= . If a« be 
 a simple root of R (x) = , the integral reduces to the form 
 
 r £ (x) dx R (x) =z(x — a t ) G (a?) 
 
 and is therefore finite. If all the roots of R(x) are simple, the most general 
 integral of the first kind will be of the form (28) , the degrees of the A'b will be 
 given by (29), and the number of arbitrary constants included in the integral is 
 therefore 
 
 2((i-D»-i)= ( ' Mt - a j (w - 1) 
 
 t=2 
 
 If a t is a Mold root of R (x) = , we have only to put l t = in the dis- 
 cussion above in order to show that the number of constants is reduced by 
 
 ^ — , which we know to be equal to the number of double points and 
 
 cusps equivalent to the multiple point a t on the curve /= . The number of 
 arbitrary constants in the most general integral of the first kind is therefore 
 
 (mn-2)(m — l) _ IJE _ 
 2 * F 
 
18 
 
 In the more general theory, based on the curve <p (xy) = of degree m in x 
 and y, the general integral of the first kind is found to be of the form 
 
 ( Q(xy)dx 
 
 i <P'v ' 
 
 where Q is of degree m — 3 in both x and y , and has the &-ple points of <p = 
 
 as (k — l)-ple points. Q has therefore * ^ — ' coefficients which are 
 
 subject to Ia t ^ ~~ ~^ conditions (a^^the number of i-ple points of <pz=zO). 
 
 Consequently, if there is no reduction in the number of these conditions, there 
 will be p and only p integrals of the first kind linearly independent. But in 
 order to show that there are only p we must show that no such reduction takes 
 place. In the case we have been considering we find not only the number of 
 the conditions but the conditions themselves. We thus at once see that there 
 is no reduction in their number, and affirm that there are p and only p 
 linearly independent integrals of the first kind. 
 
 VI. 
 
 Periods of Integrals of the First Kind. Their Form, and 
 Reductions in their Number. 
 
 If we ask after the number and character of the periods of these integrals 
 of the first kind, we can proceed in two ways. We can, in the first place, form 
 the Rieman surface connected with the curve /=0. It will be m sheeted and 
 of genus p. We know that by bending and stretching it can be deformed into 
 a sphere* with p handles,! and therefore that the integral of a uniform function 
 has on it 2p periods. 
 
 In general these 2p periods are distinct, i. e. there exists between them 
 no linear homogeneous relation with integer coefficients. The proof of this 
 last statement need not be given here ; as it is the same as in the case of the 
 general Abelian Functions ; except that, in the consideration of the inequality 
 
 \ XdY >0,t 
 
 * Jordan, Liouville's Journal, Series 2, t. XI (1866). 
 
 f Klein. On Rieman's Theory of Functions. Section 8 of Miss Hardcastle's transla- 
 tion. 
 
 JPicard, Traits d' Analyse, Tome 2, pp. 405-409. X-\-iY is an integral of the first 
 kind. iT is the contour made up of the p cuts through the holes, the p cuts around the 
 holes, and the p — 1 cuts joining these into a continuous contour. 
 
19 
 
 we note that, in the region of the branch points, -=- is no longer of the order 
 - — , but of the order ^-r in the case of a simple branch point : and of the order 
 
 »7~i (* <C m ) m * ne case °f a multiple branch point, (r is the radius of the 
 
 circle of integration about the branch point.) 
 
 We might proceed to form a system of normal integrals and to discuss the 
 relations existing among their periods : but the discussion is so strictly analogous 
 to that usually given for the general Abelian Functions that we prefer to turn 
 to the system of integrals already formed, and consider the form, and certain 
 reductions in the number, of their periods. 
 
 We assume, first, that all the factors of R (x) == are distinct. Take upon 
 the plane an arbitrary point u , and draw from it loops to all the mn branch 
 points of / (i. e. the roots of R (x) =: 0) . Denote the loop around a t by a] 
 when it is described in the positive direction, and by of* when it is described 
 in the negative direction. 
 
 ■2niS 
 
 Take now any one of our integrals of the first kind 
 
 ^r = [ <p(x)dx ^ 
 J my 8 
 
 The effect on Us of a small circle around a t is to multiply it by e m ==>* 
 The value of U along the straight line from u to the small circle about a t 
 will be denoted by A t . 
 
 The inverse function y will have as periods the values of U s along any 
 contours which return y to its original value. Among such contours we 
 choose the following, 
 
 «1 «2 > a l «3 9 a l « 4 , . . . . «i a mn , 
 
20 
 
 and denote the corresponding periods by 
 
 co 2 , w s , . . . , w mn . 
 Consider the first of these. We have evidently 
 
 ( o 2 =:A 1 {l—^ m - 1 ^)+A 1 (^- 1 ^ — ^ m -^) + +A 1 (X*—l)+A 2 (X—l). 
 
 If now we multiply this by X s , we have a new period X s w 2 to which corresponds 
 the contour af -2 ^^. In the same way we have X 28 co 2 corresponding to 
 af _8 «2«i, etc. Treating the other co's in the same way, we have the following 
 table of periods associated with Us , 
 
 co 2 , X s w 2y X™<o 2 , X( m -V & a) 2 , 
 
 co 5 , X s co 3 , X 28 w 3 , ^ m " V s co 3 , 
 
 «i 4 , X s co i} X 28 co 4 , X {m - 1)8 w if 
 
 <»mn , X S 0) mn , X 28 CO mn , . . . X {t 
 
 ■1)8, 
 
 (30) 
 
 We say farther that this table of periods is complete, i. e. that all possible 
 periods associated with Us can be expressed as linear homogeneous functions, 
 with integer coefficients, of the periods (30). 
 
 In order to show this we note : 
 
 1° The most general period of y corresponds to a contour made up of 
 km -\- 1 loops described in the same direction, and / loops described in the 
 opposite direction, where h and I may have any positive integer values including 
 zero. (The same loop described r times is regarded as r loops.) 
 
 2° Between any two loops of any period we may introduce the nugatory 
 contour afcq~ k , corresponding to the period . 
 
 3° Given a general period w x and its corresponding contour, there exists 
 a period X u co x , whose contour consists of the same loops arranged in the same 
 cyclic order as in the contour belonging to co x , but beginning at an arbitrary 
 point in the cycle. (This is merely a generalization of what has already been 
 done in the case of w 2 , X s w 2 , .... X^ m - 1 ^w 2 .) 
 
 4° The new periods, — a) t + co k , izfzlc, corresponding to the new 
 contours af l a k ; together with the period w 2> which corresponds to both the 
 contours «f _1 «;> and aj^a^y enable us to replace the last loop of any contour by 
 any other loop. 
 
 1°, 2°, 3°, and 4° being granted, let it be required to form from the 
 periods (30) an arbitrary period corresponding to a contour, denoted by 2), 
 consisting of km + 1 positive and / negative loops. (The argument will 
 
21 
 
 be the same for hm-\-l negative and I positive loops.) To do this repeat the 
 contour corresponding to co 2 k times ; and from this, by the introduction of the 
 proper nugatory contours «f«7"*> we g e * a new contour A which has the same 
 number of positive and negative loops as D and in the same order. By 
 successive cyclic permutation of the loops of A ; and the addition, after each 
 permutation, of the proper one of the periods deduced in 4°; A becomes 
 identical with D ; and the corresponding period is given as a linear homogeneous 
 function, with integer coefficients, of the periods (30). The system (30) is 
 therefore complete. 
 
 There are, however, certain reductions among the periods (30). We 
 know that the value of Us along a contour composed of all the loops is zero. 
 But this contour, by a process entirely analogous to that used in the case of D 
 and A y may be reduced to the contour corresponding to io 2 repeated n — 1 
 times and followed by w i . Moreover, the periods used in this reduction are 
 none of them derived from o) t . We may therefore express co t in terms of the 
 other periods, and strike out from the table 30 the row of periods 
 
 a) iy X 8 o) if X 2S o) iy A<« 
 
 1)8, 
 
 The remaining mn — 2 periods in the first column of (30) are in general 
 distinct, since each of them contains an A that does not appear in any of the 
 others. 
 
 If d is prime to m , the table (30) , by virtue of the relation x m = 1 , and 
 by a permutation of the columns, takes the form 
 
 > 2 , Xco 2 , Fw 2 , t m Jo>2, 
 
 >s , *co 3 > ^3 f . . . . A m ^3 , 
 
 (0 mn-l> ^ w mn-\y * ^mn-1 y • • • «* m ^r 
 
 (31) 
 
 (We have chosen the last row as the one to be dropped.) If m is odd, we have 
 between the periods of any row the one relation 
 
 k = m — 1 
 Oh 
 
 >£p = 
 
 fc=0 
 
 We may therefore strike out any column (say the last). The remaining periods 
 are in general distinct; and the integral has, under the hypotheses made 
 above as to d and m, the maximum number of independent periods, i. e. 
 
 (mn — 2) (m — 1) = 2p . 
 
22 
 
 If m is even (d still prime to m) we have the relation 
 
 /12- = _1 
 
 A 2 = — A«. 
 
 ?/i 
 
 When -g is odd, we can express all of the even powers of X in terms of the odd 
 powers; and the table of periods (31) may be replaced by the table 
 
 Xto 2 , 
 Xco z , 
 
 X z co 2y 
 Xho B , 
 
 .•.'.* 
 
 CO 
 
 i) 
 
 ~) 
 
 ^°mn-li A C0 mn _ ly 
 
 (32) 
 
 m, 
 
 But X is now a primitive — th root of unity, and therefore 
 
 2 
 
 x+x B +x 5 + +>- i =yA»+i == o 1 
 
 k = 
 
 We may therefore drop the last column of (32), and the integral has now only 
 (mn — 2) I — — 1 ) periods, which are in general distinct. (We note that this is 
 
 m 
 
 twice the genus of the curve y* — R /m-v 2n (a;) ■=. , all the factors of B being 
 distinct.) 
 
 If, on the other hand, - is even, we can drop half the columns of (31) ; but 
 
 we know of no relation connecting those which remain. The integral has in this 
 
 case (mn — 2) =- periods which are in general distinct. 
 
 If d is not prime to m , put S=zs/j. and m = ?y* , where s is prime to r . 
 We have then the relation X rS = 1 , and from this the relation 
 
 k=m—'l k = r — l 
 
 2>l M = ;/£>>. 
 
 fc=0 fc=0 
 
 The table of periods (30) accordingly takes the form 
 
 co 2 , X s co 2 , X 28 co 2 , ^"^Vo 1 
 
 ':;> 
 
 X S (D* 
 
 \0 2 
 
 «>mn-l, ^V 
 
 mn — 1 f 
 
 >a> 
 
 mn — 1 > 
 
 . ..AC- 1 * 
 
 . . Ji'- 1 )** 
 
 'mn — 1 • J 
 
 (33) 
 
«F^ 
 
 2niS 
 
 But /I 5 = e"»»~ = e" = /5 8 , where /? is a primitive rth root of unity. Since r is 
 prime to s and /9 r = 1 , (33) may, by use of this value of ^ and proper permu- 
 tation of the columns, be put in the form 
 
 co 2 
 
 CO 
 
 3> 
 
 (0 mn — 1 
 k=r — 1 
 
 
 
 >{>< 
 
 'mn — 1) 
 
 F 
 
 fr- x m. 
 
 mn — 1 
 
 (34) 
 
 When r is odd, 2J* k ~ , we strike out the last column, and have in general 
 
 fc = 
 
 (mn — 2)(r — 1) independent periods. (We note that this is twice the genus 
 of the curve y 2 — B rt , n (x) = , all the factors of B (x) being distinct.) 
 
 If r is even, a discussion exactly analogous to that made in the case when 
 
 m was even and d prime to m shows that we have in this case (mn — 2) f -^ — l) 
 
 or (mn — 2) independent periods, according as ^ is odd or even. 
 We may tabulate all these results as follows : 
 
 m odd, the no. of periods is (mn — 2)(m — 1). 
 
 d prime to m 
 and 
 
 d = sfi. m = r/u. 
 
 s prime to r 
 
 and 
 
 odd, 
 
 even, 
 
 r odd, " 
 "oodd, « 
 
 2 even, 
 
 (mri _2)g_l). 
 
 (mn-2)g). 
 
 (mn— 2)(r— 1). 
 ( TO »_2)g-l). 
 
 (mn-2)(0. 
 
 The meaning of a portion of these reductions is very evident. When 
 d=zs/jt and m=.r/jt y the integral 
 
 
 x^dx 
 
 m(R mn (x)} 
 
 m y 
 
 x^dx 
 
 s ' 
 
 m (B mn (x))' r 
 
 which last is an integral connected with the curve y r — B mn (x)=zO. I am able 
 at present to offer no satisfactory explanations of the other reductions. We 
 
24 
 
 have said that the periods, after the above reductions have been made, are 
 in general distinct. It is evident that the only farther reductions which 
 can arise, so long as the factors of R(x) are distinct, must come from relations 
 among the a»'s themselves. It may be possible to so choose the roots of 
 R (x) = that some at least of the integrals connected with the curve shall 
 have less than ran — 2 distinct co's. For example, the integral of the first 
 kind, 
 
 [x n ~Hx 
 
 connected with the curve 
 
 y m — (x n —a^)(x n — a^) (x n — a%) = 
 
 reduces to the integral 
 
 [dx 
 
 connected with the curve 
 
 y rn_^_ a n )(x _ Q n ) _ (*_<) = 0, 
 
 when we take x 11 as our new variable ; and the new integral has only ra — 2 w's. 
 A similar case is the reduction of the integrals connected with the sextic 
 
 y^ — (x i — al)(x 2 — ai)(x 2 ~(4) = 
 
 to elliptic integrals.* 
 
 It is evident, however, that no farther reductions can take place among 
 the periods derived from anyone co , so long as the factors of R(x) are distinct. 
 Therefore, while in the case of the hyperelliptic integrals we can only say that 
 there are at least two periods, we are able in the present case to say that the 
 integral 
 
 (xPdx d=zs/i 
 
 has at least (ran — 2) ( ^ — 1 ) distinct periods. 
 
 We have limited ourselves so far to the case where all the factors of R (x) 
 are distinct. The case where some of them are the same presents no insuper- 
 able difficulties, but introduces a great deal of complexity. We shall limit 
 ourselves to a simple case. 
 
 Suppose h (k<^m) of the factors of R (x) to be of the form (x — a } ), the 
 others remaining distinct. We shall have in this case mn — 2 — h co's corre- 
 
 *Picard, Traits' d' Analyse, Tome I, p. 217. 
 
25 
 
 sponding to contours of the form «f _1 «o and one, coj , corresponding to a 
 contour of the form aj* _1 «f . If k is prime to ra , the point a t will accordingly 
 give rise to m — 1 periods and we have in all (ran — k — l)(ra — l)=z2p 
 periods. 
 
 If k is not prime to ra , put k = lp and m-=.Xp , and by the argument 
 used when d was not prime to ra we see that the point ctj gives rise to X — 1 
 periods, and we have in all (ran — k — 2) (ra — 1) + ^ — 1 « 
 
 The number of periods is in this case therefore (/> — 1)(X — 1) less than the 
 maximum 2p . The number of periods in both these cases will of course be 
 subject to reduction when ra is even or when d is not prime to ra . 
 
 We have been speaking so far of curves reduced to the standard form 
 
 y m — B mn (x)=zO, 
 
 but similar relations exist among the periods associated with the curve, 
 
 y m —R s {x) = 0, 
 
 where R is rational and entire and s is any integer. The difference in the 
 discussion will arise from the fact that the point infinity is now a branch point 
 where all the values of y permute in one or more cycles. We will have then a 
 similar complete system of periods co t , and their multiples by rath roots of 
 unity. 
 
 In particular, if we make ra zz: 3 and s = 4 , we have for the three integrals 
 connected with the curve y z z=.x(x — a)(x — b)(x — t) , the periods given by 
 Picard,* 
 
 ho[, Xco{ f , ho[", 
 
 
 
 Comptes Rendus, Tome 93, p. 835. 
 
Biographical Sketch. 
 
 The author, William H. Maltbie, was born in Toledo, Ohio, August 26, 
 1867. He was under the care of private instructors and in various elemen- 
 tary and high schools until 1885, at which time he entered the Ohio Wesleyan 
 University at Delaware, Ohio, from which institution he received the degree of 
 A. B. in 1890, and of A. M. in 1892. In 1890 he was appointed Professor of 
 Mathematics in Hedding College at Abingdon, 111. In 1891 he entered the 
 Johns Hopkins University as a candidate for the degree of Doctor of Philo- 
 sophy, selecting Mathematics as his principal subject, with Astronomy as first 
 and Physics as second subordinate. In January, 1893, he was appointed 
 University Scholar, and in June, 1894, Fellow in Mathematics. 
 
 un rv 
 
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