73-75] Solution in Powers of a Parameter 75 



no constant of integration being added since w l must vanish when X = 0. 

 Integrating equations (190), (191)... in turn, we find 





 w (n) _ _ \ _ 1 _ nn+i p 



{(n + l)\}*^ L 

 If we suppose </> expanded in powers of the parameter e in the form 



< = efa + e 2 (/> 2 + e s (f> 3 + ... 

 we have 



= P - (i/) P + (i/) 2 D 2 P - 2 (i/) 3 ^ + 



75. To evaluate terms in e 2 , we proceed to equations (186) and (187). 

 For brevity, we shall limit our discussion to that particular type of distortion 

 which ultimately proves to be of importance for the problem immediately in 

 hand. For this, as will appear in the next chapter, u^ is of degree 3 in 

 f, rj, f. Equations (184) and (185) accordingly . shew that w l must be of 

 degree unity, and w^ must vanish. Similarly, u 2 will be found to be of 

 degree 4, so that w 2 is of degree 2, w 2 ' of degree zero and w" = 0. Again u s 

 will be of degree 5, w 3 of degree 3, w s ' of degree unity, w 3 " = 0, and so on. 



The value of v 1 is accordingly 



f 1 = -iDP = -i(AP {f + BP,, + CP ff ) ............ (193). 



Proceeding to the determination of second order terms, we have from 

 equations (182) and (177), 



= - i (APf + BP,* + CP^ 2 ) + Q - iP (AP + HP,,, + CP) 



= -pP 2 + Q ......................................................... (194). 



The value of w 2 can next be found from equation (186). The right-hand 

 member reduces to V 2 w 2 , and the equation, expressed in f, 77, f coordinates, 

 becomes 



so that w 2 = -^D 2 P 2 - \DQ ........................ (195), 



while similarly equation (136) leads to 



