INVERSE METHOD OF DEFINITE INTEGRALS. 321 



We must except, as before, from the application of this theorem 

 the case where V is of the form {tt')-\Vi, and r greater than, or 

 equal to unity. 



5. If (f) be any of the transcendants contained in the indefinite 

 integral jj (tt')", where m is between — 1 and + x exclusive, and if 



^"~ 1.2.3...ndt"'^"^ ' 

 then shall Qn be a self-reciprocal function for integrations relative to <p. 



For Q„ is evidently of the form — ~rp: ■ ~TZ' ^'^d ^ is not 



of the form excepted in Art. 3., since m is between —1 and + oo. 

 Moreover, by actual differentiation we get 



1 .^.S-.-ndt" 

 where a, b, c, &c. are constant quantities. 



Hence, 



Q„ = at"' ^btt'"-' +ctH"'-^ + kc., 



which is of m dimensions in t, and therefore all the conditions re- 

 quired in Art. 3. are here fulfilled; therefore Q„ is a self-reciprocal 

 function relative to <p. 



6. If (p be any of the transcendants expressed by the indefinite 

 integral jj (tt')"", where m is between + 1 and — oo exclusive, and if 



qn = 



- d° (tt')°- 



1.2.3....ndt' 



n» ■ 



then is qn a self-reciprocal function relative to (p. 



d' (tfy V 

 For §-„ is here of the form i ' — , and V does not belong to 



the excepted cases, moreover 



# _ d'^.jtt'y-'" 



^'' dt ~ \.2...ndt"-^^^^ 

 is evidently of n dimensions in t, therefore all the conditions of 

 Art. 4. are here satisfied. 



