INTEGRATION OF DIFFERENTIAL EQUATIONS. 101 



The second line is equivalent to 



jpg j^cos (qx + a.) - ; jcos fe# + a) --sin faaj + a) 



2 2 ) 



- - sin (qx + a) -- 8 cos (qx + a) K 



CC <7iC j 



and thus 



/ / 1C 



?/ = 0-jsin (go; + a) H -- cos (^a? + a) 5-5 sin (^a? + a) 



[ O'iC O' X 



15 



These examples will sufficiently illustrate the general formula. 

 The same method is applicable to the equation 



Here we have 



n[(n-i)(n - 2) -p (p - 1)} a n + fa n _, = 0, 



Let 



and generally 



n (n -p - 1 + v) (n +^ - 2 - z/) ? + g 8 ^ = ...... (24), 



where v is divisible by 3. 

 If p is so too, let p = v, 



.'. w ^> l + v = w 1 and n + p 2 v = w 2. 

 If p 1 is divisible by 3, let j? 1 = z/, 



/. -^ - 1 + v = - 2 and ^-2-z/ = -l; 

 and in both cases (24) becomes 



w ( w _l ) ( n _ 2 )Z M + < ? X- 3 = 0, 

 and therefore ^,l n x n fulfils the equation 



= ........................ (25). 



