Abbildung, conforme 11
Abel 456
Abelian transcendental functions, arising by inversion of functions of the first kind on a Riemann's surface 455
Abelian transcendental functions, Weierstrass's form of 456
Absoluter Betrag 3
Accidental singularities 16 53 64
Accidental singularities, are isolated points 65
Accidental singularities, form of function in vicinity of 64
Accidental singularities, if at infinity and there be no other singularity, the function is algebraical polynomial 69
Accidental singularities, if there be a finite number of, and no essential singularity, the uniform function is rational, algebraical and meromorphic 71
Accidental singularities, must be possessed by uniform function 64
Accidental singularities, number of, in an area 67 68 72
Accidental singularity 16 53
Addition-theorem, algebraical 297
Addition-theorem, algebraical, is possessed by algebraical functions 297
Addition-theorem, algebraical, is possessed by doubly-periodic functions 299
Addition-theorem, algebraical, is possessed by simply-periodic functions 298
Addition-theorem, condition that algebraical equation between three variables should express 310
Addition-theorem, definition of algebraical 297
Addition-theorem, for uneven doubly-periodic function of second order and second class 247
Addition-theorem, for Weierstrass's  -function 262
Addition-theorem, form of, when function is uniform 311
Addition-theorem, function which possesses an algebraical, is either (i) algebraical 300
Addition-theorem, function which possesses an algebraical, is or (ii) simply-periodic 303 305
Addition-theorem, function which possesses an algebraical, is or (iii) doubly-periodic 307
Addition-theorem, function which possesses an algebraical, is or (iii) doubly-periodic, satisfies a differential equation between itself and its first derivative 308
Addition-theorem, partial form of, for the  -function and the  -function 261
Addition-theorem, reference to binomial differential equations 490
Adelphic order 317
Algebraic (multiform) functions defined by algebraical equation 161
Algebraic (multiform) functions defined by algebraical equation, branch-points of 162 168
Algebraic (multiform) functions defined by algebraical equation, cyclical arrangements of branches round a branch-point 171
Algebraic (multiform) functions defined by algebraical equation, in connection, with Riemann's surface 338
Algebraic (multiform) functions defined by algebraical equation, infinities of, are singularities of the coefficients 163
Algebraic (multiform) functions defined by algebraical equation, infinities of, are singularities of the coefficients, graphical method for determination of order of 164
Algebraic (multiform) functions defined by algebraical equation, when all the branch-points are simple 174
Algebraic equation between three variables should express an addition-theorem, condition that 310
Algebraic equation, class of 349
Algebraic equation, defining algebraic multiform functions 161 see
Algebraic equation, for any uniform function of position on a Riemann's surface 371
Algebraic function on a Riemann's surface, integrals of 387
Algebraic function on a Riemann's surface, integrals of, branch-points of integrals of 393
Algebraic function on a Riemann's surface, integrals of, infinities of integrals of 390 393
Algebraic function on a Riemann's surface, integrals of, integrals of, everywhere finite 388
Algebraic function on a Riemann's surface, integrals of, integrals of, everywhere finite, number of, in a special case 388
Algebraic function on a Riemann's surface, integrals of, when all branch-points are simple, three kinds of integrals of 389
Algebraic function, rational integral 70
Algebraic functions on a Riemann's surface, constructed from normal elementary functions of second kind 457
Algebraic functions on a Riemann's surface, determine a fundamental equation for a given Riemann's surface 462
Algebraic functions on a Riemann's surface, relations between zeros and infinities of 468
Algebraic functions on a Riemann's surface, Riemann — Roch's theorem on 459
Algebraic functions on a Riemann's surface, smallest number of arbitrary infinities to render this construction possible 457
Algebraic functions on a Riemann's surface, smallest number of infinities of, which, except at them, is everywhere uniform and continuous 460
Algebraic functions on a Riemann's surface, which arise as first derivatives of functions of first kind 460
Algebraic functions on a Riemann's surface, which arise as first derivatives of functions of first kind, are infinite only at branch-points 460
Algebraic functions on a Riemann's surface, which arise as first derivatives of functions of first kind, determined by finite zeros 462
Algebraic functions on a Riemann's surface, which arise as first derivatives of functions of first kind, most general form of 461
Algebraic functions on a Riemann's surface, which arise as first derivatives of functions of first kind, number of infinities of, and zeros of 461
Algebraic isothermal curves, families of 576 see
Algebraic relation between functions automorphic for the same infinite group 653
Algebraic relation between functions automorphic for the same infinite group, class of, in general 654
Algebraical addition-theorem 297
Algebraical function determined by an equation 161
Algebraical function, rational 70
Amplitude 3
Analytic function, monogenic 56
Analytical curve 409 423 530
Analytical curve, area bounded by, represented on a half-plane 530
Analytical curve, area bounded by, represented on a half-plane, consecutive curve can be chosen at will 531
Analytical curve, represented on a circle 423
Analytical test of a branch-point 157
Anchor-ring conformally represented on plane 501
Anharmonic function, automorphic for the anharmonic group 620
Anharmonic group of linear substitutions 620
Anissimoff 111
Appell 145 295 296 464
Appell's factorial functions 464 see
Area, bounded by analytical curve represented on half-plane 530
Area, bounded by cardioid on half-plane 536
Area, bounded by circular arcs 549 see
Area, form of function for representation on a circle 529
Area, form of function for representation on a plane 528 540
Area, of convex rectilinear polygon 537 see
Area, simply connected, can be represented conformally upon a circle with unique correspondence of points, by Riemann's theorem 526
Areas, combination of, in proof of existence-theorem 425
Argand 2
Argument 3
Argument (or amplitude) of the variable 3
Argument and parameter of normal elementary function of third kind 453
Argument and parameter, interchange of 451
Argument of function possessing an addition-theorem, forms of, for a value of the function 300
Arithmetic mean, method of the 408
Ascoli 409
Ausserwesentliche singulaere Stelle 53
Automorphic function 619
Automorphic function, connection between, and general linear differential equations of second order 656
Automorphic function, constructed for infinite group in pseudo-automorphic form 638 see
Automorphic function, different, for same group are connected by algebraical equation 653
Automorphic function, different, for same group are connected by algebraical equation, class of this algebraical equation in general 654
Automorphic function, expressed as quotient of two thetafuchsian functions 651
Automorphic function, its essential singularities 651
Automorphic function, modular-functions as examples of 657
Automorphic function, number of irreducible zeros of, is the same as the number of irreducible accidental singularities 651
Automorphic functions 582 619
Barriers, can be used to classify connected surfaces 314
Barriers, changed into a cut 314
Barriers, impassable, in connected surface 313
Beltrami 530 532 533
Betrag, absoluter 3
Bianchi 617 618
Biehler 295
Bien defini 161
Biermann 55 297
Bifacial surface 325
Bifacial Surfaces 325 333
Binomial differential equations of first order when the integral is uniform, with the various classes of integrals 482
Bolza 583
Bonnet 500
Borchardt 218
Boundaries of connected surface, relation between number of, and connectivity 324
Boundary 322
Boundary conditions for potential function 410 see
Boundary of region of continuity of a function is composed of the singularities of the function 57
Boundary values of potential function for a circle 414
Boundary values of potential function for a circle, include all the maxima and the minima of a potential function 422
Boundary values of potential function for a circle, may have limited number of finite discontinuities 417
Boundary, assigned to every connected surface 314 322 329
Boundary, defined 322
Boundary, edges acquired by cross-cut and loop-cut 315
Boundary, effect of cross-cut on 323
Boundary, effect of cross-cut on, and of loop-cut on 324
Boundary, of simply connected surface is a single line 323
Branch 15
Branch-line 339
Branch-lines, are mode of junction of the sheets of Riemann's surfaces 339
Branch-lines, free ends of, are branch-points 340
Branch-lines, number of, when branch-points are simple 364
Branch-lines, properties of 340
Branch-lines, sequence along, how affected by branchpoints 341
Branch-lines, special form of, for two-sheeted surface 344
Branch-lines, special form of, for two-sheeted surface, when all branch-points are simple 356
Branch-lines, system of, for a surface 341
Branch-point 15 154
Branch-points, analytical test of 157
Branch-points, are free ends of branch-lines 340
Branch-points, canonical arrangement of, when simple 364
Branch-points, circuits round two, are irreducible 350
Branch-points, defined 15 154
Branch-points, deformation of circuit on Riemann's surface over, is impossible 350
Branch-points, effect of, on branches 149 151
| Branch-points, effect of, on sequence of interchange along branch-lines 341
Branch-points, expression of branches of a function in vicinity of 158
Branch-points, in connection with loops 357 see
Branch-points, integral of a function round any curve containing all the 37
Branch-points, joined by branch-lines when simple 344
Branch-points, number of simple 175
Branch-points, number of, when simple 356
Branch-points, of algebraic functions 162 168
Branch-points, simple 174 355
Branch-section 339
Branches of a function, a function which has a limited number of, is a root of an algebraic equation 175
Branches of a function, affected by branch-points 151
Branches of a function, analytical expression of, in vicinity of branch-point 158
Branches of a function, are uniform in continuous regions where branch-points do not occur 155
Branches of a function, defined 15
Branches of a function, number of, considered 159
Branches of a function, obtained by continuation 151
Branches of a function, of an algebraic function 161 see
Branches of a function, restored after number of descriptions of circuit round branch-point 157
Branches of a function, which are affected by a branch-point, can be arranged in cycles 156
Brill 356 367 464
Brioschi 275 281
Briot 456 464
Briot and Bouquet vii 23 39 41 168 173 208 210 218 230 456 482 489 490
Burnside (W.) 117 247 355 406 524 536 558 620 640 654
Burnside (W.S.) and Panton 390
Canonical form of complete system of simple loops 361
Canonical form of complete system of simple loops, Riemann's surface 365
Canonical form of complete system of simple loops, Riemann's surface, resolved 366
Canonical resolution of Riemann's surface 355
Canonical resolution of surface 355
Cantor 147
Cardioid, area bounded by, on a circle 536
Cardioid, area bounded by, represented on strip of plane 535
Casorati 2 23 359
Categories of corners 592 see
Categories of corners, cycles 592 596
Cathcart 6
Cauchy v vii 23 27 43 51 61 180 312
Cauchy's theorem on the integration of a holo-morphic function round a simple curve 23
Cauchy's theorem on the integration of a holo-morphic function round a simple curve and of a meromorphic function 27
Cauchy's theorem on the integration of a holo-morphic function round a simple curve on the expansion of a function in the vicinity of an ordinary point 43
Cayley 2 11 74 356 476 482 509 510 515 529 530 533 537 549 575 576 579 620 622
Cesaro 92
Christoffel 538 541 549
Chrystal vi 2 6 48 170 184
Circle, areas of curves represented on area of, any circle, by properly chosen linear substitution 514
Circle, areas of curves represented on area of, any simply connected area, by Riemann's theorem 526
Circle, areas of curves represented on area of, exterior of ellipse 501
Circle, areas of curves represented on area of, exterior of parabola 505
Circle, areas of curves represented on area of, exterior of square 545
Circle, areas of curves represented on area of, half-plane 506
Circle, areas of curves represented on area of, infinitely long strip of plane 508
Circle, areas of curves represented on area of, interior of cardioid 536
Circle, areas of curves represented on area of, interior of ellipse 504
Circle, areas of curves represented on area of, interior of parabola 506
Circle, areas of curves represented on area of, interior of rectangle 502 544
Circle, areas of curves represented on area of, interior of regular polygon 548 (Ex.)
Circle, areas of curves represented on area of, interior of semicircle 507
Circle, areas of curves represented on area of, interior of square 503 545
Circle, discriminating 111
Circuit 327
Circuits, cannot be deformed over a branch-point on a Riemann's surface 350
Circuits, complete system of, contains unique number of circuits 328
Circuits, drawn on a simply connected surface are reducible 329
Circuits, number in complete system for multiply connected surface 330
Circuits, on connected surface 327
Circuits, reducible, irreducible, simple, multiple, compound, reconcileable 327
Circuits, represented algebraically 328
Circuits, restore initial branch after number of descriptions 157
Circuits, round branch-point, effect of, on branch of a function 153 155
Circular functions obtained, by integrating algebraical functions 191
Circular functions obtained, on a Riemann's surface 380
Class (of connected surface) 324
Class of doubly-periodic function of second order 223
Class of equation 349
Class of group 608
Class of singularity 147
Class of tertiary-periodic function 288
Class of transcendental integral functions 89
Class of, algebraic equation associated with a Riemann's surface 349
Class of, algebraic equation associated with a Riemann's surface, between automorphic functions 654
Class of, connected surface 324
Class of, essential singularity 147
Class of, Fuchsian group 608
Class of, Laguerre's criterion of 91
Class of, Riemann's surface 349
Class of, simple function of given class 91
Class of, tertiary-periodic function, negative 291
Class of, tertiary-periodic function, positive 288
Class of, transcendental integral function, as defined by its zeros 89
Classes of doubly-periodic functions of the second order are two 223
Clebsch 174 209 356 359 361 363 364 367 403 455 456
Clifford 333 361
Closed cycles of corners in normal polygon for division of plane 596 see
Combination of areas 425
Combination of areas, in determination of potential function 425
Complex variable defined 1
Complex variable defined, represented on a plane 2
Complex variable defined, represented on a plane, and on Neumann's sphere 4
Compound circuit 327
Compound circuits 327
Conditions that one complex variable be a function of another 7
Conformal representation 11
Conformal representation of planes, established by functional relation between variables 11
Conformal representation of planes, examples of 501
Conformal representation of planes, magnification in 11
Conformal representation of planes, most general form of relation that secures, is relation between complex variables 496
Conformal representation of planes, used in Schwarz's proof of existence-theorem 423
Conformal representation of surfaces is secured by relation between complex variables in the most general manner 492
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, form of function for, on a circle 529
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, form of function for, on a plane 528
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, obtained by making one a plane 495
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, of anchor-ring 501
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, of ellipsoid 500
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, of oblate spheroid 500
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, of sphere on plane 497
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, of sphere on plane, Mercator's and stereographic projection 498
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, of surfaces of revolution on plane 496
Conformal representation of surfaces is secured by relation between complex variables in the most general manner, Riemann's general theorem on 526
Conforme Abbildung 11
Congruent figures 517 591
Congruent regions by linear substitutions 517 591
Conjugate edges 592
Conjugate edges of a region 592 see
Connected surface 312
Connected surface, can be deformed into any other connected surface of the same connectivity having the same number of boundaries, if both be bifacial 334
Connected surface, divided into polygons, Lhuilier's theorem on 325
Connected surface, geometrical and physical deformation of 333
Connected surface, Klein's normal form of 334
Connected surface, supposed to have a boundary 314 322 329
Connected surface, to be bifacial 325
Connection of surfaces, affected by cross-cuts 319
Connection of surfaces, affected by cross-cuts, and by slit 321
Connection of surfaces, affected by cross-cuts, by loop-cuts 320
Connection of surfaces, defined 312
Connection of surfaces, multiple 314
Connection of surfaces, multiple, definition of 315
Connection of surfaces, simple 313
Connection of surfaces, simple, definition of 315
Connection, order of 317
Connectivity 317
Connectivity, affected by cross-cuts 319
Connectivity, affected by cross-cuts, by loop-cuts 320
Connectivity, affected by cross-cuts, by slit 321
Connectivity, in relation to irreducible circuits 330
Connectivity, of a Riemann's surface, with one boundary 347
Connectivity, of a Riemann's surface, with one boundary, with several boundaries 350
Connectivity, of spherical surface with holes 321
Connectivity, of surface defined 317
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