INVERSE METHOD OF DEFINITE INTEGRALS. 389 



Analytical Table of Reference to the "Memoirs on the Inverse 

 Method of Definite Integrals." 



FIRST MEMOIR, Vol. IV. Page 353, &c. 



PAGE 



Introduction 353 



Section I. Principles relative to Continuous Functions. 



Art. 1. Method of reducing the given limits of integration to and 1 in all cases 358 



Arts. 2, 3, 4. In the general equation ft f{t) .t''= <p(x), x is understood to lie between 

 — 1 and + 00 , then cyj (x) converges to zero as x increases, when y(<) is any of the 

 functions usually received in analysis; consequent division of the subject 35g 



Art. 5. Rule; When the known function <p{x) is rational, seek the coefficient of - in 



X 



<p (x) . t~', dividing it by / we obtain fit) 362 



Art. 6. Examples s6S 



Arts. 7, 8. Means of facilitating the Calculus oi f{t) 3Q5 . 



Art. 9. and Note (A). When {x) is a logarithmic function 366, 400 



Art. 10. When (pix) is expressed by an equation to finite differences 367 



Art. 1 1 . When <f> (x) is a fraction, the denominator containing imaginary factors 369 



Art. 12. When {x) is irrational 37O 



Art. 13. Cases when equations of the form J',f(J,').(t'' ±t~')=if>x, may be resolved by 



the preceding method 37] 



Art. 14. Extension of the general rule to successive integration with respect to any 



number of variables , 373 



Section II. Principles relative to Discontinuous Functions. 



Art. 15. Cases of discontinuity in Physical Problems quoted 374 



Art. 16. To find a formula which shall represent the least of the two quantities a, /3. . 375 

 Art. 18. To find a formula which shall represent_/(a) or y(/3) according as a is < or > /3. 376 



Art. 19. To find a formula which shall represent r^, or -= — j— , according as a is 



a — lip p — na 



< or > /3 377 



Art. 20. To find a formula which shall represent ■~^^ , or ^^^, , according as a is < or > /3 378 



Arts. 21, 22. Method of representing discontinuous functions of any number of breaks 380 



Arts. 23, 24. Geometrical Illustrations of the theory of discontinuity 382 



3E2 



