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THE CALCULUS

FOR

ENGINEERS AND PHYSICISTS.

BY THE SAME AUTHOR.
MEASUREMENT CONVERSIONS (English and French). 43

Graphic Tables or Diagrams, showing at a glance the Mutual Conversion
of Measurements in Different Units of Lengths, Areas, Volumes, Weights,
Stresses, Densities, Quantities of Work, Horse Powers, Temperatures, etc.
For the use of Engineers, Surveyors, Architects, and Contractors. By Prof.

R. H. SMITH. 4to. Price 7s. 6d.
"The work is INVALUABLE"-Colliery Guardian.
“Ought to be in EVERY office where even occasional conversions are required.

. Prof. SMITH'S TABLES form very EXCELLENT CHECKS on results. A VERY USEFUL and good set of diagrams."-Electrical Review.

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MANUALS INDISPENSABLE FOR THE ENGINEER.

By W. J. MACQUORN RANKINE, C.E., LL.D., F.R.S.,
Late Reg. Prof. of Civil Engineering in the University of Glasgow.

In Crown 8vo, Cloth.
Thoroughly Revised by W. J. MILLAR, C.E.,
Late Sec. to the Inst. of Engineers and Shipbuilders in Scotland.

RANKINE'S APPLIED MECHANICS. FOURTEENTH EDITION. 12s. 6d.
RANKINE'S CIVIL ENGINEERING. NINETEENTH EDITION. 16s.
RANKINE'S STEAM ENGINE AND OTHER PRIME MOVERS. FOURTEENTH

EDITION. 12s. 6d.
RANKINE'S MACHINERY AND MILLWORK. SEVENTH EDITION. 12s. 6d.
RANKINE'S USEFUL RULES FOR ENGINEERS. With Appendix for Electricians

by Prof. A. JAMIESON, M.I.El.E. SEVENTH EDITION. 10s. 6d. RANKINE'S MECHANICAL TEXT-BOOK : A Simple Introduction to the Study

of Mechanics. FOURTH EDITION. 9s.
PROFESSOR RANKINE'S MISCELLANEOUS SCIENTIFIC PAPERS. With

Memoir by Prof. TAIT, M.A., and Fine Portrait on Steel. Royal 8vo,
Handsome Cloth. 31s. 6d.

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APPLIED MECHANICS

Prof. JAMIESON. 7s. 60. BRIDGE CONSTRUCTION

Prof. FIDLER, 30s. DESIGN OF STRUCTURES . S. ANGLIN. 16s. TRAVERSE TABLES

R. L. GURDEN. 21s. MARINE ENGINEERING

A. E. SEATON. 21s. THE STEAM ENGINE.

Prof. JAMIESON. 8s. 60. CHEMISTRY FOR ENGINEERS. BLOUNT & BLOXAM. 10s. 6u. GAS, OIL, AND AIR-ENGINES . BRYAN DONKIN. 25s. BOILER CONSTRUCTION

T. W. TRAILL. 12s.6d. HYDRAULIC MACHINERY. Prof. ROBINSON. 34s. ELECTRICAL POCKET-BOOK MUNRO & JAMIESON. 8s. 6d. ELECTRICAL PRICE-BOOK H. J. DOWSING, 8s. 60. MARINE ENGINEERS' POCKET- SEATON AND ROUNTHWAITE. BOOK

8s. 6d.

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*** For other STANDARD WORKS see Technical Catalogue

at the end of this volume. LONDON : CHARLES GRIFFIN & Co., Ltd., Exeter St., Strand.

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INTEGRATION AND DIFFERENTIATION,
WITH APPLICATIONS TO TECHNICAL PROBLEMS;

WITH

CLASSIFIED REFERENCE TABLES OF INTEGRALS

AND METHODS OF INTEGRATION.

BY

Henny
PROF. ROBERT H. SMITH,
ASSOC. M.I.C.E., M.I.M.E., M.I.EL. E., M.FED.I.MIN.E., WH. SCH., MEM. ORDER MEIJI.

WITH DIAGRAMS.

LONDON:
CHARLES GRIFFIN AND COMPANY, LTD.
EXETER STREET, STRAND,

1897.

[All rights reserved.]

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PREFACE.

This work aims at the presentation of two leading features in the study and application of the higher mathematics. In the first place, the development of the rationale of the subject is based on essentially concrete conceptions, and no appeal is made to what may be termed rational imagination extending beyond the limits of man's actual physical and physiological experience. Thus no use is anywhere made of series of infinite numbers of things or of infinitely small quantities. The author believes that the logical development is both sound and complete without reference to these ideas.

In the second place, a set of Eleven Classified Tables of Integrals and Methods of Integration have been arranged in such manner as seemed best adapted to facilitate rapid reference, and thus relieve the mind engaged in practical mathematical work of the burden of memorising a great mass of formulas. This part of the work has involved very considerable labour. The germ of it is twenty-five years old in the author's manuscripts, but for the extensive and able development of it to its present form he is indebted to the cooperation of Mr R. F. Muirhead, M.A., B.Sc. (Glasgow), B.A. (Cambridge), formerly Clark Fellow of Glasgow University and Lecturer on Mathematics at Mason College.

It is hoped that these Tables may prove of great service to physicists and engineers engaged in new applications of science. In the preparation of these Tables assistance has been obtained from

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