THE APPLICATIONS OF QUATERNIONS IN DIFFERENTIAL EQUATIONS. 715 



then 



X = (D, + ^)_-»Y 2 . 

 D".(D 2 + </» 1 )- n Y 2 e 1 t *=Ye** 

 (D 2 + 2 ) " "YgCj** = D - n . Ye** 

 where, as X was arbitrary, Y may be any function of t. Similarly 



/(D) e «*X=/(D 2 + ^) ei '*X 2 

 also 



/(D)X=/(D 2 + ^ 1 )(X e -^e 1 ** 

 =/(D 2 + ^ 1 )e 1 W'(«-«*X) 2 . 



Theorem III. — (Forsyth, § 34). — 



Theorem IV. — (Forsyth, § 35). — 



f(tV)rx = t"'f(t~D + l H)x. 

 Similar theorems can be formulated when the independent variable is a linear vector 

 function with constant axes, but they are not of sufficient importance to merit a separate 

 discussion. 



For instance 



f(<f>D)<f> m X = </>"7'((£D + m)X 



where D = -=- . 

 d<p 



Properties of the General Linear Differential Equation. 



9. The general linear differential equation of the n tu rank is of the form 



tip = (<f> D" + <A 1 D«- 1 + </> 2 D- 2 . . . </>„) P = <f>a . . . (11) 



where <p . . . cp n (p are functions of t but not of p, and a is a constant vector. 

 If jOj is any particular integral, putting p = a- + p 1 we find immediately 



Then, if the general solution of this equation is <r=,<r 1 , that of (11) is p = v 1 + p y . 



If k independent particular integrals of (11) can be obtained, the order of the 

 equation can be reduced by k (see Jordan, Cows d' Analyse, i. iii., 1887, p. 138). 

 The most important case is where k = 3. In this case let the three particular integrals 

 be a, ^8, and y, and put 



p = aSicr + ftSjo- + ySka = xj/cr 



then 



Clp={<f> il,D n + (<l> 1 t + n<t> Dt)D"-' 1 ••• +Jty}o- = <£a . . (12) 



in which, as Q^ = 0, we may regard Do- as the dependent variable. 



The second term of (12) may be removed by solving the linear equation 



