INVERSE METHOD OF DEFINITE INTEGRALS. 391 



PAGE 



Art. 15. General principle for finding Reciprocal Functions to simple integration 130 



Art. l6. The same extended to integration for any number of variables 131 



Art. 17. Examples 132 



Section V. Inverse Method for Junctions which contain positive ponters of x, or are 

 under any other form. 



Art. 18. An appendage must be annexed in all such cases 135 



Arts. 19, 20. When ^j; is a rational and entire function of x ; and particular example 



when ^(.r) = l 136 



Art. 21. To find /(<) when7(/(O-''=0W, an<i ^ is from to n—\ inclusive 138 



Arts. 22, 23. Various modes of determining y(<) in this case 141 



h 



Arts. 24, 25. The coefficient of h" in the expansion of , is a self-reciprocal function 146 



THIRD MEMOIR. Vol. V. Page 315, &c. 

 Introduction 315 



Section VI. Method of discovering Reciprocal Functions, when the integrations are 



performed with respect to any fonction of the variable. 



Arts. 1, 2. General principle for varying the limits 318 



d'.^t'f'V) dt 

 Art. 3. If V can be found so that — "' — 'Tl "™^y "'^ ''^ " dimensions in t (where 



t' = \ — t) then this quantity will be self-reciprocal relative to (p $iq 



Art. 4. If V can be found so that — j^ ■ j? "^^y be of n dimensions in t, then 



the factor by which -j^ is multiplied will be self-reciprocal relative to 320 



Arts. 5, 6. If <p=f,(Jt(y indefinite, and m between —1 and -j-oo, and if 



d^ {(tt'Y'^"'\ 

 Qn = 1 2 xdf-^" ' ("')""" *^" *^^^ ^ ^^ self-reciprocal relative to (p 321 



n — m 



rhfti'\ 



If z= ftittfy indefinite, and m be between -1-1 and —00, and if a» = — ' 



^ -"^ ' > ' 1" i.2...ndt'" 



then shall q, be self-reciprocal relative to 321 



Art. 7. To find the functions which Q„, 5, generate 322 



Arts. 8, 9. When 7« = — §, Q„, q, are the trigonometrical reciprocals 323, 325 



Art 10. In the identities thus obtained, the sign of n may be changed so as to pass 



from differential coefficients to integrals 326 



Art. 1 1 . The two series of reciprocal functions obtained from the theorems of Arts. 5 81. 6. 



differ only with respect to the variable of integration 328 



Art. 12. Examples of the preceding theory 329 



Art. 13. To express Q„ and q„ in terms of t alone 330 



Art. 14. To express Q„ and q„ by means of differential equations 332 



