Definite Integrals, with Physical Applications. 369 



introduce imaginary quantities, when decomposed into simple 

 factors, may more conveniently be left in the quadratic form ; 

 as in the following example. 



Ex.8. Given »(*) = frlffifp - 

 It is easily seen that 



and therefore by Ex. 2. 



/3>(h.l.i)' ft h. 1.(1)' 



+ ( B -^V.,|^.^i)' +&c j 

 + l~ 1.2.3 +&C -j 



«*'-*k cos (/3h.l. )) + ^j*- .sin (/3b.l. i)J 

 put m s = ^ 2 + ( J "^ " ) and 7 = tan- . B _ Aa . 



Hence /(0 - *"" ' • m sin (7 + j8 h. 1. -) ; 



which result if differentiated n times with respect to /3" as inde- 

 pendent variable, will solve the case of » equal quadratic- 

 factors. 



