THE APPLICATIONS OF QUATERNIONS IN DIFFERENTIAL EQUATIONS. 719 



15. When & is readily factorisable, the following method may be used : — 

 Let 



Qp=(D-^)(D-^)(D-e) . . . (D-(9)p = CTa 



.-. (D - MD - c) .. . . (D - 6)p= jj^^a + e f ff3 

 (D - e) ... (D - 6)p = (D - f )-HD - ^-^o 



= (D - ^) " '(D - </>) " 'cJa + (^ - f ,) _ * e\fi 



and so on. 



If 0\|^ ... are commutative we have evidently 



p = 2e'*a + n~'CTo . 



16. Homogeneous Equations — (Forsyth, § 55). The equation 



fip = (</> CTCT n D' ! + < p 1 CTCT"- 1 D"- 1 • • • +<p„)p = . . . (20) 



where ct is a variable linear vector function with constant axes, can be reduced to the 

 case of constant coefficients by changing the dependent variable to xs and then putting 



p = T3<r and w = e 9 , which gives -j- =e~ e — 7 . and therefore 



cto civ 



ilxs" \<I0 J\il6 J " ' \'W JdQ 



The more general substitution p = e m9 <r cr = e 6 is sometimes useful. 



17. Exact Differential Equations — (Forsyth, §§ 56, 57). — The condition of exactness 

 is found as in ordinary differential equations to be 



Linear Equation of the Second Rank with Variable Coefficients. 



18. The general form is 



<PoP + 'f>iP + <l>-2P = <t>"- 

 where (f> <p l (f> 2 and <p are variable linear vector functions. 



If three independent particular solutions a , fi , y of the subsidiary equation are 

 known, the equation can be solved (cf. Forsyth, § 58). 

 For putting 



p = (<xS . i + /JS .j + yS . k)<x = T3<t 



we get 



<f> n zzv + (2cp ti + <f> l iz)& + (<f> i3 + <p 1 zz + $. 2 &)a = cpa . . (21) 



<p £W + (2<p £T + <p l T3)& = <pa 



which is linear in &. 



