39. Advanced Calculus, III: Definite Integrals.-Theory and applications, in connection with the elements of the theory of elliptic functions. Mj. Summer Quarter, PROFESSOR SHAW.

45. Critical Review of Secondary Mathematics. (For Teachers.) A brief survey of the subject-matter of Secondary Mathematics as seen in the light of modern mathematics, aiming both to organize the theory of the whole scientifically and to gather the products of this work for use in teaching. Prerequisite: working knowledge of Secondary Mathematics. Mj. Summer Quarter, ASSOCIATE PROFESSOR YOUNG.

46. Graphical Methods in Algebra.-(For Teachers.) The cross section paper as a mathematical instrument. M. First Term, Summer Quarter, PROFESSOR MOORE.

GRADUATE COURSES

NOTE. Students should register for graduate courses only after consultation with instructors.

65A, B, C, D. Reading and Research in Pure Mathematics.—When in the judgment of the Department it is advisable that students undertake definite mathematical reading and research not closely connected with any current lecture course or seminar, this work will be directed by PROFESSOR MOORE, PROFESSOR BOLZA. ASSOCIATE PROFESSOR BLISS, OR ASSOCIATE PROFESSOR DICKSON. Mj or DMj. Each quarter.

66A, B, C. Reading and Research in Applied Mathematics.-Advisable reading and research will be directed by PROFESSOR MOORE, PROFESSOR MYERS, OR DR. LUNN. Mj or DMj. Each quarter.

71A, B. Theory of Numbers: A, Theory of Congruences; B, Theory of Algebraic Numbers.-Introductory courses, the second presupposing the first. A: M. First Term; B: M. Second Term, Summer Quarter, AssociATE PROFESSOR DICKSON.

71A. Elements of Theory of Numbers.-Mj. Autumn Quarter, PROFESSOR MOORE.

74A, B. Theory of Algebraic Numbers.-In particular, Hensel's exhibition of analogies with Weierstrass' theory of Functions. A: Mj. Winter Quarter; B: Mj. Spring Quarter, PROFESSOR MOORE.

84. Continuous Groups.-An illumination of the fundamental concepts and theorems of the Lie theory in connection with various classes of problems of Geometry. ASSOCIATE PROFESSOR DICKSON. [Not given in 1909-10.]

85. Differential Equations from the Standpoint of Lie.- Prerequisite: course 39. ASSOCIATE PROFESSOR BLISS. [Not given in 1909-10.]

94. Theory of Invariants.-The theory of binary quantics ("modern higher Algebra") with applications to the theory of equations and to Geometry. Prerequisite: courses 32 and 39. ASSOCIATE PROFESSOR DICKSON. [Not given in 1909-10.]

112A, B, C. Introduction to General Analysis, I, II, III.—A study of the type of General Analysis in which at least one of the independent variables of the theory enters without direct characterization as to quality or range of variation, being, for the simplest instances (1) a single fixed element; (2) an element having a fixed finite number of values; (3) an element having a numerable infinitude of values; (4) an element with values corresponding to the real number from 0 to 1, inclusive-the general theory applying to these and all other instances in which the postulated body of functional relations is verified. The course presupposes a good knowledge of the elements of the theory of functions of real variables, and is intended to lead to independent investigations in various chapters of general analysis. Prerequisite: course 38, and a certain general mathematical maturity. Three consecutive half majors. Autumn, Winter, and Spring Quarters, PROFESSOR MOORE.

112A. Introduction to General Analysis.-Prerequisite: course 38 and a certain general mathematical maturity. M. First Term, Summer Quarter, PROFESSOR MOORE.

115A, B, C. Integral Equations in General Analysis.-A development of the theory of Fredholm, Hilbert, and others in the sense of General Analysis. Prerequisite: course 112. Three consecutive half majors. Autumn, Winter, and Spring Quarters, PROFESSOR MOORE.

121. Theory of Functions of a Complex Variable: Cauchy's Theory.— Introduction to the Algebra and Calculus of complex numbers and their geometric representation; conform representation and Cauchy's theory of functions. Prerequisite: courses 37, 38. DR. LUNN.

122. Functions of a Complex Variable: Weierstrass' Theory.-Reading, problems, and lectures. Fundamental theorems on limits and continuity; convergence of series and operations upon them; studies of special functions -algebraic functions and the more usual transcendental ones. Text: Harkness and Morley's Introduction to Analytic Functions. Prerequisite: courses 37, 38, and 39. Mj. Summer Quarter, ASSISTANT PROFESSOR KELLOGG. Mj. Autumn Quarter, PROFESSOR Bolza.

123. Weierstrass' Theory of Elliptic Functions.-Prerequisite: course 122. Mj. Winter Quarter, PROFESSOR BOLZA.

124. Applications of Elliptic Functions.-Especially to problems of Geometry and Mechanics. Prerequisite: course 123. Mj. Spring Quarter, PROFESSOR BOLZA.

126. Linear Differential Equations.-The general theory with particular reference to Differential Equations of second order. Special study of the hypergeometric function. Prerequisite: course 121. ASSOCIATE PROFESSOR DICKSON. [Not given in 1909-10.]

127. Modern Theories of Analytic Differential Equations with Applications to Celestial Mechanics.-3Mj. Autumn, Winter, and Spring Quarters, ASSOCIATE PROFESSOR MOULTON.

128A, B. Partial Differential Equations.-Especially the linear equations. Geometrical interpretations. Applications to Geometry and Physics. Prerequisite: course 151. Two consecutive half majors. Winter and Spring Quarters, AsSOCIATE PROFESSOR BLISS.

131. Synthetic Projective Geometry.-Determination of fundamental postulates; theory of one dimensional system of first and second order. M. First Term, Summer Quarter, PROFESSOR Moore.

141. Modern Analytic Geometry. -Homogeneous (trilinear) co-ordinates; projective properties of conics. Prerequisite: courses 31, 38, and 39. [Not given in 1909-10.]

142. Higher Plane Curves.-General properties of algebraic curves. Special study of curves of the third and fourth order. Prerequisite: course 141. [Not given in 1909-10.]

151. Differential Geometry.-Theory of twisted curves and surfaces. Prerequisite: Advanced Calculus. Mj. Autumn Quarter, ASSOCIATE PROFESSOR BLISS.

160. Analytic Mechanics, I.-(Astronomy 5.) Dynamics of a particle. Prerequisite: courses 26, 27, and 31. Mj. Summer Quarter; Mj. Autumn Quarter, AssOCIATE PROFESSOR LAVES.

160 A. Analytic Mechanics, II.-(Astronomy 6.) Prerequisite: course 160. Mj. Winter Quarter, ASSOCIATE PROFESSOR LAVES.

161. Vector Analysis.-The elements of vector algebra, vector differentiation and integration, and the linear vector function; illustrated by typical applications to geometry, mechanics, and physics. Prerequisite: courses 31 and 39. Mj. Summer Quarter, DR. LUNN.

161A, B, C. Vector Analysis.-Three consecutive half majors. Autumn, Winter, and Spring Quarters, Dr. Lunn.

162. Theory of Attraction and the Potential. The potential function of gravitation and electrostatics; Laplace's equation, Green's functions, and harmonic analysis; extensions to cases of heterogeneous media, with sketch of the abstract theory as related to linear differential equations of the second order. Prerequisite: courses 39 and 160A. DR. LUNN. [Not given in 1909-10.]

163. Differential Equations of Mathematical Physics.-A study of certain typical linear partial differential equations occurring in the analytic representation of physical theories, and of the functional expansions of solutions satisfying given boundary conditions; with special reference to the concrete phenomena and physical analogies from which the abstract theory has been generalized. Prerequisite: courses 39 and 160A. DR. LUNN. [Not given in 1909-10.]

181, 182, 183. General Seminar. For the consideration of reports of current research and of literature, especially of a fundamental or critical nature. Autumn, Winter, and Spring Quarters, PROFESSOR MOORE.

PRE-ENGINEERING COURSES

Certain courses offered in the University constitute parts of the curriculum in all schools of engineering, and may be pursued with advantage by those planning to enter a school of engineering later. The branches of Engineering for which preparation is offered include Civil, Mechanical, Mining, Chemical, and Electrical Engineering. The courses announced elsewhere in this circular, which form part of the curriculum of Engineering, are the following: Mathematics, courses 1, 2, 3, 18, 19, (20 recommended); Physics 3, 4, 5; Chemistry 28, 3S, 6, and courses in English, History, and Modern Languages similar to those required of all students in science. These courses may be taken by students preparing for all branches of Engineering. Courses in Drawing, Surveying, and Shop-work, described below, are also open to intending engineers.

Tuition.-All students registered for any of the technical courses mentioned below, except Introduction to Surveying, pay the usual tuition of $40 per quarter, and an additional $10 for each major of technical work. Students paying Engineering tuition, if their scholarship entitles them to carry four courses, may take three college courses in addition to one technical course; but students are advised to take only two college courses in addition to one technical course.

COURSES OF INSTRUCTION

1. Freehand Drawing. The work of this quarter consists in developing the powers of observation, the appreciation of proportion, harmony of form and color. Pencil, charcoal, color, and pen and ink are used as media. Freehand projection drawings of various details of building and cabinet construction, together with their isometric and perspective sketches; freehand projections, with their isometric and perspective sketches, of the conventional solids; light-and-shade studies, in black and white, of various solids, casts, vases, etc.; color and pen-and-ink studies of assigned subjects. A series of home-sketches covering the subjects taught is required. Estimated time for required work, outside of classroom periods, 15-30 hours. Required of students in all branches of Engineering. Lectures and classroom work, 10 hours a week (60 hours). M. First Term, Autumn Quarter, MR. FERSON.

2. Descriptive Geometry and Mechanical Drawing, I.-Use and care of instruments, scales, triangles, etc.; the graphic solution of problems in plane geometry (text, Faunce, Mechanical Drawing); elementary lettering; titles, dimensioning. Six drawings required. Estimated time for required work, outside of classroom periods, 15-30 hours. Required of students in all branches of Engineering. Lectures and classroom work, 10 hours a week (60 hours). Prerequisite: course 1. M. Second Term, Autumn Quarter, MR. FERSON.

3. Descriptive Geometry and Mechanical Drawing, II.-Orthogonal projection of points, lines, planes, and solids; plans, elevations, and sections; projections in first and third angles; application of geometry to problems in machine design. Interpenetrations; their development and their application to machine design. Isometric and cabinet projection. Shade lines, titles, dimensioning. Ten drawings required. Estimated time for required work, outside of classroom periods, 30-60 hours. Required of students in all branches of Engineering. Lectures and classroom work, 10 hours a week (120 hours). Prerequisite: course 2. Winter Quarter, MR. FERSON.

4. Descriptive Geometry and Mechanical Drawing, III.-Pure descriptive geometry (text, Church). Projections upon right and oblique planes; intersection of lines, planes, and solids; application of descriptive geometry to problems in elementary machine design. Ten drawings required. Required of students in all branches of Engineering. Estimated time for required work, outside of classroom periods, 30-60 hours. Lectures and classroom work, 10 hours a week (120 hours). Prerequisite: course 3. Spring Quarter, MR. FERSON.

5. Mechanical Engineering Drawing.—Conventional sections for material; details of machines from given data and formulae; proportions for bolts, nuts, screw-threads, springs, pipe-couplings, shaft-couplings, journals, etc. (text, Unwin, Elements of Machine Design, Part I). Practice in arrangement of material upon the sheet, titles, dimensioning, lettering, and tabular arrangement of data. Freehand sketches and measurements of details of machines, with their office drawings, tracings, and blue-prints. Ten drawings required. Required of students in Mechanical and Electrical Engineering, and recom. mended to all others. Estimated time for required work, outside of classroom periods, 30-60 hours. Lectures and classroom work, 10 hours a week (120 hours). Prerequisite: course 3. Autumn Quarter, MR. FERSON.

6. Mechanism.-Problems in belts and pulleys, based upon given data and formulae; gear teeth under the cycloid and involute systems; bevel gears and elliptical gears (text, A Treatise on Gear Wheels). Cams and their graphics; the plain slidevalve; measurement sketches; assembled drawings, tracings, and blue-prints of some machine or parts. Required of students in Mechanical and Electrical Engineering, and recommended to all others. Ten drawings required. Estimated time outside of classroom periods, 30-60 hours. Lectures and classroom work, 10 hours a week (120 hours). Prerequisite: course 5. Winter Quarter, MR. FERSON.

7. Topographical Drawing.-Practice in ornamental titles, borders, scales, the conventional signs used in topographical drawing; the drawing of contour maps, sections; color work applied to map-drawing; computation of areas, volumes, etc., for excavations and fillings. Ten drawings required. The required work may require time outside of classroom periods. Required of all students in Civil and Mining Engineering. Classroom work, 10 hours a week. Hours: 2:00-5:00. Prerequisite: Mathematics 1, Engineering 4. Mj. Winter Quarter, MR. FERSON.

8. Surveying (Course 5, Department of Mathematics).-Field-work with chain, tape, compass, transit, and level (text, Breed and Hosmer, The Principles and Practice of Surveying). Required of students in Civil and Mining Engineering. Prerequisite: Mathematics 1, and course 7 above. Spring Quarter, DR. MACMILLAN.

9. Forge and Machine Shop-Work.-Forge: Study of tools and materials used. Instruction in the building and management of fires, heating, bending, twisting, upsetting, drawing out, welding of iron and steel, tempering of steel. Machine-shop: Study of tools and machinery used; chipping, filing, scraping, fitting, centering, chucking, turning, reaming, finishing, polishing, drilling, tapping, screw-cutting. A certain amount of work will be required of all students, and opportunity for advanced work will be offered to the more ambitious and skilful men. Recommended to mechanical, mining, chemical, and electrical engineers. 10 hours a week. Prerequisite: course 4. Spring Quarter, MESSRS.

XVIII. THE DEPARTMENT OF ASTRONOMY

AND ASTROPHYSICS

OFFICERS OF INSTRUCTION

EDWIN BRANT FROST, A.M., Professor of Astrophysics, and Director of the Yerkes Observatory.

SHERBURNE WESLEY BURNHAM, A.M., Professor of Practical Astronomy, and Astronomer in the Yerkes Observatory.

EDWARD EMERSON Barnard, A.M., Sc.D., LL.D., Professor of Practical Astronomy, and Astronomer in the Yerkes Observatory.

GEORGE ELLERY HALE, S.B., Sc.D., LL.D., Non-resident Professor of Astrophysics (Mt. Wilson, Cal.).

KURT LAVES, A.M., PH.D., Associate Professor of Astronomy.

FOREST RAY MOULTON, A.B., PH.D., Associate Professor of Astronomy.

JOHN ADELBERT PARKHURST, S.M., Instructor in Practical Astronomy at the Yerkes Observatory.

STORRS BARROWS BARRETT, A.B., Secretary and Librarian of the Yerkes Observatory.

PHILIP FOX, S.B., Instructor in Astrophysics at the Yerkes Observatory. WILLIAM DUNCAN MACMILLAN, A.M., PH.D., Instructor in Astronomy.

DANIEL BUCHANAN, A.B., A.M.

FELLOW, 1909-10

INSTRUCTIONAL WORK

The work of the Department of Astronomy and Astrophysics is divided into two parts: (1) Work at the University, comprising: (a) elementary instruction in general Astronomy, both theoretical and practical; (b) preliminary training in the principles and methods of work underlying the science of Astrophysics (given in part in the Department of Physics); (c) graduate and research work in Celestial Mechanics. (2) Graduate and research work in Practical Astronomy and Astrophysics in the Yerkes Observatory at Lake Geneva.

In the work at the University, given by Associate Professors Laves and Moulton and Dr. MacMillan, emphasis will be laid on the development of the mathematical principles and methods which form the basis of the mathematical sciences. In addition to the courses in Descriptive Astronomy, Introduction to Celestial Mechanics, and Analytical Mechanics, courses in the various branches of Celestial Mechanics will be given within periods not