328 Mr MURPHY'S THIRD MEMOIR ON THE 



Dividing the last equation by {tty, and integrating witt respect 

 to t, we get 



1.3.5...(2w-l) d-'^^-'iffy-i 1 . c i/i o.M 

 7 — x; — ^ • TT — 1 = - Sin \ncos-Ul-2t)}. 



Putting » = ???- 1 , we get 



, ,, 1.3.b...(2m-S) <?-"(«') -'" + -i . <,, , ,,, ^,, 



('"-1) • {-2)'"-' • dt^ "''" {(i-»i) cos-' (l-2t)}, 



thus are obtained the corresponding formulae for negative indices. 



11. The two series of reciprocal functions arising from the theorems 

 in Arts. 5. and 6., differ essentially, only in reference to the inde- 

 pendant variable of integration, for in Art. 5., ni may be any quantity 

 between —1, and +x, and in Art. 6. any quantity between +1 and 

 — 00 ; change in the latter theorem m into — m, and the limits of w 

 Avill then be the same in both ; for distinctness, also let 6 be used 

 instead of (p in the value of §'„. 



d" itt'Y*'" 

 Hence, Q^= i ^.s.'.ndf ■^*^'^'""' ^"'^ <t> = !^itfY, 



d" (ttY'^" 



^- lALndt" ^ ^"d ^ = ;(«')-'". 



Now the reciprocal functions of Art. 5., give the equation 

 UQnQn=0, or feQM,. ^=0. 



But ^ =(«')", and ^ =(«')-'" ; therefore ^ = («7'». 



Hence, feQAtt'Y ^ QAtt'Y = 0. 



And since QAtt'T = qn, and QAtt'y = q„; it follows that UQnQn' is 

 equivalent to [dq„qn; the only difference being with respect to the 

 variables (f> and 9 employed for integration. 



