CONTENTS 
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CHAPTER XVII 
THE CALCULUS OF VARIATIONS AND HAMILTON’S PRINCIPLE 
PAGE 
b 
Maximum or Minimum of J* F(x, y, y')dx . . ■ . 
a 
Euler’s Equation 
Minimum Surface of Revolution 
The Brachystochrone 
Definition of the Variations 
Euler’s Equation for Multiple Integrals 
Laplace’s Equation in Curvilinear Coordinates. Dirichlet’s Prin 
ciple .....••••••• 
Isoperimetric Problems 
Variable End Points 
Parametric Form and the so-called “Variation of the Inde 
pendent Variable 
Hamilton’s Principle ......... 
Least Action ........... 
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CHAPTER XVIII 
THERMODYNAMICS. ENTROPY 
Reversible Changes and the (v, p) - Diagram ..... 449 
The First Law of Thermodynamics 452 
Differentials ........... 453 
In Particular, the Differential dQ . . . . . . . 455 
The Entropy of a Perfect Gas ....... 457 
CHAPTER XIX 
DEFINITE INTEGRALS AND THE GAMMA FUNCTION 
The Definite Integral as a Function of a Parameter. Leibniz’s 
Rule 460 
Several Parameters and Multiple Integrals ..... 463 
Improper Integrals.......... 464 
Tests for Convergence ......... 466 
Absolutely Convergent Integrals ....... 468 
The Limit Tests .......... 469 
Alternating Integrals ......... 472 
Infinite Integrands .......... 474 
Continuation ........... 477 
Discontinuities within the Interval ...... 479 
The Gamma Function ......... 480 
The Beta Function ......... 484 
Improper Double Integrals 485 
Evaluation of Definite Integrals by Differentiation . . . 486 
Other Methods 488