INVERSE METHOD OF DEFINITE INTEGRALS. 329 



If in the formulEe of Arts. 5. and 6., we assign to m all possible 

 values between —1 and 4-1, we obtain two series of self-reciprocal 

 functions, which when m = become identical with each other, and 

 with the functions denominated P„ in the preceding memoirs. For 

 every other value of m between those limits, there are two different 

 kinds of reciprocal functions, one of which only is a rational and entire 

 function of t, for instance when m= —\, we have found the functions 

 cos n(p and sin {7i + l)<p, the former of which only is a rational fvmc- 

 tion of cos <p. 



12. (1.) W/ien m= -i- 



To determine cp in this case, make sin 9 = ii — / ', squaring and ob- 

 serving that t + f' = l, we get sin^ 6 = 1 -2 {tt')K whence » 



/■i + ?;'i = V^(2 - sin- 0), and 2'\tty = cose. 

 Differentiate the assumed equation, and we get 



^os ^ = a ^**'\\ • ~7^ ' therefore — -^ . -r^ = 2 cos . -7 



;i 5 



2 {tty • cie' ^"-'-— ^tt')i ■ d0~ ■ti + t'i 



hence, (p = 2E {e)-F(d). 



The extreme values of the amplitude of these elliptic functions 



being — -, and + -; the limits of ^ are 0, and 4:Ei — 2JF\, where 



El and Ei denote the complete functions when the amplitude extends 

 from zero to a right angle. 



The reciprocal functions for integrations relative to (p, are 



_>3.7.11...(4?i-l) 



Q,= 



4.8.12 4.W 



4n{^n-l) 4n{4>n-l)(in-i){4u-5) „ , 



^* sTi * ^^ 3.4.7.8 ^ (,^^-U 



