SECT. I.] IDENTITY OF SOLUTIONS. 367 



we shall give it the following form 



Idq e~ q * sin x cos 2q *Jkt + jdy e~^ cos x sin 2q ^ki ; 

 or else, 



,P/ fe-M e -t 



4- / da e~i cos x ft . _ ^ f _ 1 

 ./-* V2V-1 2V- I/ 



which is equivalent to 



e-** sin x (jdq e -(9- v -*0 2 + i /^ e -(&amp;lt;/+ V-w&amp;gt;A 



4- e-* cos a? 



the integral ]dq ***=** taken from ? = - x to ^ = x is V^ 



we have therefore for the value of the integral (dqe-* sin (#+2? i/ kt), 

 the quantity VTT e~ w sin a?,- and in general 



VTT er n2 *&amp;lt; sin w^ = J ^ e~^ sin n(x + 2q V^) , 

 we could determine in the same manner the integral 



,+ 



I c?2 e-3 3 cos n (x + 2^ ^S) , 

 the value of which is V? e ^ 1 cos ?i#. 

 We see by this that the integral 

 e-W (a, sin n.a? + \ cos w.a?) + e~ n * ki (a, sin w 8 a; + 6 2 cos w^) 



+ e-&quot;&quot; 2 ^ (a a sin w 3 ic -f 6 3 cos up) -f &c. 

 is equivalent to 



-i Cdq 9 ~* I* 1 Sin Wl (iC + 2 2 V ^) + a a sin w, ( 4- 2^ VS) + &c. | 



v/7rj. (^ cos Wl (a: + 2j V^) + 6 8 cos 7i a (x 4- 2 2 V^) 4- &cj 



