to ascertaining by the polar system of any plane figure from given data-Different problems relating to a cylinder of any section whatever, and to forces distributed over a plane area-Centre of gravity of a surface and volume of revolution-Complement to the rules of Guldinus—Centre of pressure. (6) Bending moment and shearing force of a straight beam loaded directly and indirectly—Relative diagrams-Reaction of supports and section of greatest danger-Line of influence of Frankel-Polygon of addition of Winkler—Theorem of Levy-Bending moment and shearing force of a solid with curved neutral axis-Diagrams and corresponding theory-Diagrams of moments and shearing forces—Continuous beams. (7) Internal forces—Simple and compound bending and graphical construction of united maximum forces-Problems for the construction of a section, satisfying given conditions, and having a given modulus of resistance and moment of inertia (8) Straight beam of constant section -Determination of the section—Variation of the united maximum forces from one extremity to the other-Straight beam of equal resistance, and determination of their form (longitudinal section) under different conditions–Elastic line and its differential equation-Theorem of Mohr --Application to straight beams on simple supports, or encastre at one or both extremities-Conditions of stability and of arches. II. DRAWING CLASS IN GRAPHICAL STATICS. Time.-Five hours weekly, for one-half year. Spiral of Archimedes, equiangular spiral, logarithmic curve, catenary, etc—Graphical exercises in products, powers, roots, logarithms, resolution of numerical equations—Measurement and division of areas and volumes-Division of field lines of boundary, etc.—Graphical calculation of movements of the earth -Lines of pressure in an arch, and tension in a catenary-Exercises in composition and resolution of forces and of couples, system of chainsConstruction for shearing force and bending moment for any section of a straight beam resting on two supports, and variable loading, fixed, moving, concentrated and uniformly distributed representive curves-Section of greatest danger, reactions of supports—Exercises on beams loaded indirectly-Constructions in tension and compressions of a jointed girder -Constructions relating to the movements of parallel forces, allied antipolar system, conic of inertia, principal axes, etc.-Centre of gravity, central ellipse, and kernels of sections used in practice, constructions and relative applications, centre of pressure, hydraulic wheels, etc.Exercises on problems of simple and compound bending-Determination of the dangerous section of a beam, given all its external forces-Construction for internal forces—Elastic line—Exercises on interpolation and least squares. APPENDIX III. DATA OF THE TEACHING OF GEOMETRY IN EUROPEAN ENGINEER ING SCHOOLS. Descriptive. Mech. & Civ. El. 1 4 Practical. Civil, Analytical, Civil & Mech. 5 Projective. Civil. 4 Descriptive. Mechanical Analytical. Civil Mech. Elec. 6 Descriptive. Clvil Mech. Elec. 6 Descriptive. Civil & Mech, 2 Descriptive. Civil & Mech. 3 win.4 sum Analytical. Civil & Mech. Analytical. Civil. 1 Analytical. Civ. Sur. Me. El. 6 Descriptive. Civ, Sur. Me. El. Projective. Civil, Surveying. Analytical. Civil & Mech. 6 Descriptive. Civil & Mech. Practical Civil. Practical. Civil. Practical, Mechanical 3 Practical. Mechanical Analytical. Civil Elec. Mech. 4 Descriptive. Civil Elec. Mech. Analytical. Civil Elec. Mech. 2 I 1 Civil & Mich. DISCUSSION. THE CHAIRMAN (Dr. H. T. Eddy) stated that the subject was a live one in engineering schools, and that it was important to know how much graphical work should be introduced into technical technical courses. He hoped the paper, which was a valuable one, would be fully discussed. PROFESSOR MANSFIELD MERRIMAN agreed with the general propositions advanced. It was his opinion that graphical methods were pretty thoroughly taught in all engineering schools in this country, but not always put down under a separate heading. Graphics is usually taught in connection with the study of roofs and bridges. In his own practice he gave seventy to eighty exercises to the study of stresses in simple trusses, of which about one-fourth were devoted to graphical methods. He urged that the diagrams should be drawn in ink and then be carefully scaled. He said that graphical methods had not come into as general use and prominence in this country as was expected when they were introduced fifteen or twenty years ago. He could not explain the reason, unless it were that the bridges were so largely of one simple type. PROFESSOR J. B. Johnson thought graphics might better be taught in a course by itself without any reference to its applications. He thought the course in geometrical drawing and descriptive geometry could be combined with graphics. PROFESSOR D. S. JACOBUS explained that in the institution with which he was connected graphical were methods used in the course in algebra and mechanics, and found to be very useful. He thought there should be a separate course in graphics aside from statics. PROFESSOR R. C. CARPENTER gave instruction in graphics in connection with the solution of all theoretical problems, but many times he found difficulty in applying graphical methods owing to the lack of drawing tools. PROFESSOR JOHN GOODMAN agreed with the author of the paper, and advocated a thorough study of graphics. Such a course is given at Leeds and at the University of London. PROFESSOR O. H. LANDRETH thought the teaching of graphics of more importance than many appreciate, but advocated teaching the principles and the applications at the same time. PROFESSOR H. W. SPANGLER believed it the best plan to teach the elements of graphics early in the course and independent of its applications. PROFESSOR J. H. KINEALY wished to know when graphics should be taught. In his opinion, graphical statics could come early in the course, but the graphics that a mechanical or an electrical engineer required must come later. PROFESSOR S. W. ROBINSON remarked that some use the word graphics improperly as synonymous with graphical statics. He emphasized the point that graphical statics is only a tool by which principles are applied, and that a knowledge of graphics by no means does away with the importance of a thorough |