344 Mr MURPHY'S THIRD MEMOIR ON THE 



jeW cos wy — -. j— g— g ^ the limits of being and ir, these formulee 



\ apply for all integer values of n, except 



Now e cos nO ± e' sin nO = e^^s* cos {x sin + «0|. 



Hence /ee"""^ cos {arsin^- w0| =7r . — — -— , 



l^^xcose cos {arsin0 + wej =0. 



The particular case where w = is included in the first of these two 

 equations. 



20. By the results thus obtained, we are enabled to represent any 

 rational and integer function of a; in a form adapted to general differen- 

 tiation. 



By applying Maclaurin's theorem, we first have 



(}>(x) = Ao + A,.x +^2- j^ + ^3 - ^ g 3 + &c.; 



and passing to definite integrals by the formulae of the last article, 



(h{x) = - /ge^cose 1^^ cos (x sin 9) + A^ cos {x sin 6-9) 



+ ^2 cos (.r sin - 20) + &c. J 

 also if A^u A^i, A_3, &c. represent arbitrary constants, 



= - /ee^'=°^* {A-i cos(x sin + 0) + ^_2 cos (ar sin + 20) 



+ ^_3 cos (a; sin + 30) + &c.} 



both of which integrals must be added before <p (x) can be subjected in 

 a complete form to general differentiation. 



We then obtain the w*'' differential coefficient by adding n9 under 

 each cosine in this sum, that is. 



