714 MR J. H. MACLAGAN-WEDDERBURN ON 



The General Linear Differential Equation with Constant Coefficients. 



7. Preliminary Formulas. — Before proceeding further, it is necessary to explain a 

 convention which allows linear vector functions to be separated freely from their 

 arguments and to be treated as scalars. AVhen a linear vector function is separated 

 from its argument a suffix is attached to both, and the former treated as a scalar. The 

 following examples define the notation and make its use obvious : — 



Y<t>p<r + Ypcpcr = (</>! + <£,) Y Pl (T. 2 

 Y<t>pcj>cr = (p^Y Pl <r 2 



<pa i//a x<x 



${3 W xP 



<py xf,y X y 



<£l '/'l Xl 



<t>2 ^2 X2 



^3 «A 3 X3 



<£l $2 ^3 



"Al ^2 ^3 



Xl X2 X3 



a 1^273 



o-iP-iYs 



As it is often convenient to denote different functions by the addition of suffixes to 

 the same symbol, care must be taken not to confuse the two uses. The meaning, how- 

 ever, is generally clear from the context. 



8. In what follows f(<p) is used to denote any function of a linear vector function <fi. 

 The coefficients mf(<p) if not constant are supposed to act subsequently to (p, i.e., f((p) 

 is of the form 



ij/cp" + xfi'- 1 • • • + Z3 + ■ ■ ■ 6<p-'" ■ ■ ■ 



where ^ cp etc. are linear vector functions. ' 



Theorem I. — (Forsyth, § 32). — 



/(D)e** =/(<£)e«+ 



(p being any constant linear vector function and D = -=- ; for 



Theorem II.— (Forsyth, § 33). — 



If X is any function of t (in general a linear vector function) we have 



/(D)(Xe**)=/(D 2 + «A 1 )X 2 e 1 ^ 



for 



so that 

 Put 



D{Xe<* } = (DX + X^)e«* = (D 2 + ^ 1 ) X 2 e i t * 

 D" { XeW} = (D 2 + ^ 1 ) M X 2 e 1 «* . 



(D 2 + ^)»X 2 = Y 



! 



