1893.] On Operators in Physical Mathematics. 511 



Partial Fractions and Normal Solutions. 



11. There is also the method of partial fractions. It is not always 

 applicable, and is especially inapplicable when the removal of' 

 boundaries drives the roots of the determinantal equation into con- 

 tiguity. But the application is very wide, nevertheless. Put 

 Z = T" x j then the solution of F = Y/, when/ is constant, starting 

 when t = 0, is 



where Z is the steady Z, got by taking p = in Z, and the summa- 

 tion ranges over the roots of the equation Z = 0, considered as an 

 algebraical equation in p. That is, p is entirely algebraical in (14). 

 Similarly, the effect of an impulse / is represented by 



and from this again, by time- integration, we can obtain an expression 

 for the effect due to any varying /, which may be quite as un- 

 manageable as the previous definite integral for the same. On the 

 other hand, (14) and (15) furnish the most direct and practical way 

 of investigating certain kinds of problems, whether there be but a 

 few or an infinite number of degrees of freedom. This method is 

 the real foundation of all formulas for the expansion of arbitrary 

 functions in series of normal functions. For, find fche impressed 

 forcive that would keep up the arbitrary state. We may then apply 

 the above to every element of the forcive to find its effect, and by 

 integration throughout the system get the arbitrary functions ex- 

 panded in normal functions. Or, without reference to impressed 

 force, find the differential equation connecting any element of the 

 initial state and the effect it produces later. It will be of a form 

 similar to our F = Y/, and it may be similarly solved by a series, 

 which contains the expression of the expansion of the initial state in 

 the proper functions. 



Or we may investigate the normal functions themselves, and em- 

 ploy their proper conjugate property to obtain the expansion repre- 

 senting any initial state. But this method does not apply very 

 naturally to equations of the form we are considering. 



Decomposition of an Operator into a Series of Wave Operators. 



12. There is also another method which contrasts remarkably with 

 the previous, viz., to decompose the operator Y into a series of other 

 operators of a certain type expressing the propagation of waves. This 

 is best illustrated by an example. Suppose the question is, given an 



