for Differential Equations of Mathematical Pliysics. 587 



Since the above expression for y may be assumed to 

 remain true for an infinite number of values of the class 

 variable, the boundary relations for each of the functions X , 

 X ly etc. can be expressed in a particularly simple form. 

 Suppose, for example, the boundary condition be that at a 

 certain point, say a?=«, 



y — a + aiC + a 2 C 2 + etc. ; 



then the boundary conditions for the functions X etc. are, 

 at # = «, 



X = « 0j Xj = « 1? etc. 



In the same way, if the differential equation be one involving 

 more than one independent variable, a similar series of 

 boundary conditions for X , X 1? etc., now functions of more 

 than one independent variable, are easily derived. 



In those cases where the equations are linear and involve 

 only one independent variable there is another method of 

 presentation which bears a close resemblance to that given 

 above. Taking an equation of the fourth order for example, 

 the general solution may be written in the form 



y = A/ 1 (*) + B/ 1 (*) + Q/i(*) + iy 1 (*) l . . (4) 



where A, B, C, and I) are four quantities to be determined 

 by the boundary conditions. Instead of, as in the previous 

 method, supposing y expanded in a power series of the 

 class variable, we may imagine each of the functions f\, f 2} 

 f&f* so expanded. If these expansions can be determined, 

 the boundary conditions when inserted will provide similar 

 expansions for A, B, C, and D. For certain cases, parti- 

 cularly with linear equations of this type, this latter form of 

 presentation is frequently the simplest. The rirst step in 

 the analysis is to determine the form of the coefficients 

 in the expansion. Unless it can be proved that a con- 

 vergent expansion of this type is always possible, it will be 

 necessary to check the result a posteriori. For illustrating 

 these points the classical engineering problem of deter- 

 mining the whirling speed of a shaft whose cross-section 

 varies along its length will be treated in detail. It will be 

 seen that all the conditions of rapidity of convergency 

 and ease of calculation are most satisfactorily fulfilled. 

 Generally the method that will be adopted will be to 

 assume an expansion for the dependent variable as a 

 power series in the class variable, to insert this in the 

 differential equation, and equate the coefficients of the 



