112, us] Rotating Cylinders 111 



The third order terms are determined by equation (327). Equating co- 

 efficients as before, we obtain the equations 



f }f e 5 + f e 3 



in which of course the unknown quantity S 2 still appears. To satisfy the 

 first of these equations we must take 



The second equation is not, as might at first have been expected, an 

 equation for e 3 . If S 2 nas a certain value, it is satisfied by any value of e s , 

 but if S 2 has a value different from this, there is no solution other than 

 e- A = x . An examination of this equation will elucidate at once the whole of 

 the difficulty that was encountered in determining the true second order 

 solution in the three-dimensional problem (cf. 93 99). 



For e 3 to have a finite value, 8 2 must have the value 



g 2= _8_6_2.5_ (332). 



The third equation now does not become an equation for ^ but for e 3 + -f-e l . 

 It is satisfied by 



e s = i6^~ + * e i = *\ 



where X may have any value. Finally, the fourth equation does not deter- 

 mine X ; it reduces merely to e^ = 0, and so merely provides a check on the 

 accuracy of our work (cf. 109). 



113. Collecting the values of the various constants, we find as the equation 

 to the surface (equation 293), 



4- 2 1- (f 4 + *? 4 ) - W (f 2 + ^ + W ) 



+ #3 ji|5 (5 + ^ _ i_7,j)7R. (3 + ^j + terms in 4 , 5 , etc. (333). 



The occurrence of the indeterminate quantity X can easily be accounted 

 for. For if we have a solution 



^ = ^+0/^6% + 0/,+ ..................... (334), ' 



corresponding to a parameter 6 which is connected with the rotation by the 

 relation 



l-r, 2 /27r /3 = S + ^^ 2 +^^ 3 + .................. (335), 



then we can obtain precisely the same solution in another form on replacing 

 the parameter 6 by 6 + X0 3 . It accordingly appears that the quantity X is 



