for Differential Equations of Mathematical Pliysics. 60i> 



this physical argument it seems justifiable to assume that 

 there is a solution for the differential equation in the neigh- 

 bourhood of C = 0, approximately given by 



where 4 n 



Let the solution therefore be written 



ir = ^o + % 



for C small, and let us assume that x is also small. Inserting 

 this in the differential equation and boundary conditions, we 



Assuming % and its derivatives are small when is small, 

 an assumption that will be checked a posteriori, we must 

 therefore write to a first approximation 



where x must satisfy the boundary condition 



~dz> 'dy 



round all the boundary, and the corresponding integral 

 conditions. The quantity x therefore represents once more 

 the deflexion of a flat plate loaded with a definite finite 

 distribution, but with density proportional to 0. It therefore 

 follows that we must write ^ = CN|r|, where t/r is finite and 

 independent of C and yfr x satisfies the differential equations 

 already used in determining ^ t in the previously assumed 

 expansion. By referring back to equation (20) it is now 

 evident that we were justified in fteglectinsj the remaining 

 terms on the right-hand side, for all the derivatives of x 

 from the flat-plate analogy are of the order ( •. This step-by- 

 step process may be continued along the same lines where we 

 find that on seeking to determine the finite function, which is a 

 solution of" the differential equation for small values of 0, the 

 power series previously assumed is derived, i/r , i|r,, ^r 2 , etc. 

 being determined from the same svstem of equations as before. 

 In point of fact, the expression for i/r so found is a Taylor's 

 series regarding- C as the variable. 



