716 



MR J. H. MACLAGAN-WEDDERBURN ON 



10. If (f> is determinate, i.e., if (p is not degenerate, we have 



(D' + tftJ?" 1 • • • * _1 ^»)p=^o"V 

 and therefore, by Cauchy's existence theorem, the order of the equivalent system 

 is Sn. 



If </> is degenerate, we may proceed as follows. Suppose, in the first instance, 

 that (p is a planar linear vector function. It can then be put in the form 



. b . (TqCTo S . 0" Q O", 



Sa^o-y 



So-^20-3 



Let \p r be defined by the equation 



\ fcxTjCTgCTg/ 



b . 0"iO"2 



(13) 



and let 



Substituting in (13) we find 



4> r = <Tjb . a,. + cr 2 b . B r + 0" 3 S . y r 

 ij/ r = o-jS . a ,. + o\ 2 S . B r + cr 3 b . y' r 





a,. = fa' r B' r = gB'r 



yr=y' r - 



Therefore 



Cip = <f>a 





is equivalent to 









^(D» + ^D" + ^D"-» . 



■• fn)p = 



together with 









°°lZl^ B "- ,+ ^"~'- 



• <An)p = 



bcr 1 (r 2 i/'a 

 3 So-jO-jO-g ' 



If the complete solutions of 



(D^ + ^D"" 1 •• ■ +^)p = ^a . . . (14) 



and 



ty 1 D"- i + ^ 2 D"-» • •• ++ H )p = il,a . . . (15) 



are p x and p 2 respectively, the complete solution of ®p = (pa is 



p = ^oPi + q- 3 « ' • 



bo-j^o-g 



The modifications, where y, etc. vanish or when cp is unidirectional, are obvious and 

 not sufficiently important to merit a separate discussion. It is important to notice 

 that this process is the same whether the coefficients are constant or variable, so that, 

 if a method can be found for deducing the order of a system with constant coefficients, 

 the same method is applicable in the case of variable coefficients if we treat the 

 independent variable as a constant. 



11. Treating D as a constant, we have identically 

 (fi 3 - m-p? + m 2 Q - m s )p = 



(16) 



