CHAP. VI.] AUXILIARY MULTIPLIERS. 305 



Supposing x = in the two preceding equations we have 



n n , / a \ in r\ f du da\ 



= C + [ u -} and = D+- r &amp;lt;r w-y-1, 

 \ xj a \fc dxj a 



thus we determine the constants C and D. Making then x = X in 

 the same equations, and supposing the integral to be taken from 

 x = to x = X, we have 



du, 



f d?u 7 fdu da\ fdu da\ f d 2 cr . 



and cr -y-. ax = - 7 a u -j-\ ( 7 a u -y- + lu -=-5 cZa7, 

 J ^ \dx dxj a \dx dx] a J dx 2 



thus we obtain the equation 



- 



m C -. { ( d?(r \xj] 1 fdu da a\ 



- -j- lo-udx = \u - r - i - u T \dx + [-r- 0- U-J-+U-) 

 k j J { dx dx ) \dx dx x/ v 



fdu dcr o-\ 



- (- r &amp;lt;T-u-j- + u-} . 

 \dx dx xj a 



p *(- 



d 2 cr \x 



317. If the quantity -^ 2 -- -r which multiplies u under the 







sign of integration in the second member were equal to the pro 

 duct of cr by a constant coefficient, the terms 



u ^f-? ) dx [ and I audx 

 dx j J 



would be collected into one, and we should obtain for the required 

 integral laudx a value which would contain only determined quan 

 tities, with no sign of integration. It remains only to equate that 

 value to zero. 



Suppose then the factor a to satisfy the differential equation of 



, 



the second order y cr + -y^ -- 4^- = in the same manner as the 

 K cix dx 



function u satisfies- the equation 



m d 2 u 1 du 



F. H. 20 



