718 MR J. H. MACLAGAN-WEDDERBURN ON 



•constant. A similar result follows if h 2 or h s respectively is zero. Therefore the order 

 of the equation is the degree of h^ h 2 h 3 in g, — i.e., the degree of m 3 in D. 



It remains to consider the following cases, first where h-^ h 2 and h z , or any two of 

 them, have one or more common factors ; second, where hj h 2 or h 3 have repeated 

 roots. 



First, suppose g = g is a common root of h x = , h 2 = , h z = , then p = c" ot u is a 

 solution of the equation, where a is an arbitrary vector. If g is a common root of 

 two only, say h x and h 2 , we get from (19) / o 1 = e 0ot (A(r l + Brr 2 ), where A and B are 

 arbitrary constants. In both cases the number of arbitrary constants is the same as 

 before. 



Secondly, if h r = has an r-ple root, we have for the same value of g x 



h . = —i = ... d -— i = . 



<ty\ dl Jl 



Therefore 



P = (? ht (A 1 + A. 2 t . . . A^- 1 )^ 



is a solution of the differential equation, the number of arbitrary constants remaining 

 the same as before. 



It only remains to notice that there is always a direction <r 1 corresponding to any 

 of the values of g x ; for suppose that when g 1 =g [} , <r l vanishes, then {g x —g ) must be a 

 factor of To-j , and as T<t 1 is arbitrary, this factor may be neglected. 

 If in (17) we take conjugates we get 



x'"h'+x" l ~ 1( f>\ ■ ■ ■ *'«=o. 



Equations of this type can therefore be solved by the above method. A similar 

 investigation can be given, when the independent variable is a linear vector function, 

 by assuming p = F(<£yJ" . 



13. To find a particular solution of Op = cpa we have 



p = -1 <£a 

 = — (fi 2 — ?W 1 Q + m. 2 )<f)a . 

 = i/^a 



The theorems enunciated in paragraph 7 are useful in this connection. 



14. The following theorem is sometimes useful.* If <p is any linear vector function 

 whose constituents are integral functions of D with constant coefficients, 



is equivalent to 



0,3 = 



if the third invariant, M 3 , of (p does not contain D. For any solution of (pOp = is 

 solution of Mtftp = 0, i.e., of Qp = 0, if M 3 does not contain D. 



* See Prof. Chrtstal, "A Fundamental Theorem regarding the Equivalence of Systems of Ordinary Linear 

 Differential Equations," Trans. E.S.E., 1895. 



