THE APPLICATIONS OF QUATERNIONS IN DIFFERENTIAL EQUATIONS. 717 



m x , m 2 , and m 3 , therefore, when expanded in terms of D, constitute a set of 

 differential invariants of &. It will be shown in the next section that the degree of m 3 

 in D is the order of the system. If m 3 =0 then & is degenerate and the order of the 

 system is the degree of m 2 in D. Similarly, if also m 2 =0, the order of the system is 

 the degree of m x . 



General Linear Equation with Constant Coefficients — (Forsyth, § 166 ; 



Tait, Proc. R.S.E., 1870). 



12. If the coefficients are constants, D is commutative with the <p's, therefore from 

 (16) w? 3 |O = if Qp = 0, i.e., any solution of Qp = 0, is also a solution of m 3 p = 0~ 



If ( 1 2 3 ) are the elements of Q (i.e., the axes and latent roots), 



Sc 



hence 



Qp = //,or, - — 2 - d - +h.,a:, 5 6 l -+lt~(r~ z , — >-^- = 



/ H |w =0 h pw =0 h p™ =0 



Scr 9 cr 3 /9 S<T s cr^p So-jO-.^p 



r 3 



o " =lh o = ?2 q =lh 



offjO-^o-j oo-jO - .,^ o<r 1 a\,a\j 



and therefore 



p = <r.- i p 1 + <r a p 3 + 0-^3. 



The following proof is perhaps more satisfactory. Assume as a solution 



p = e'x a 

 where a is arbitrary and x * s a constant linear vector function. Differentiating and 

 substituting in &p = , we get 



Wdf + fcx"" 1 •••*>«0 • • • (17) 



If I x 3 [ are the elements of ^, we get immediately 

 I Pi P-i P-i ' 



(f,;/; + 4wT x + + ^»)pi = • • (is) 



or say 



© Pl = 0. 



The form of (18) is independent of the suffix attached to p and g, showing that (17) is 

 equivalent to only three scalar equations, not nine as might have been expected. 



If \ 1 2 3 I are the elements of 0, (18) is equivalent to 

 1 <r x cr 2 <r g j 



/i-,S<r 2 cT(,p ;i = h. 2 8(r s (r 1 p 1 = h s Sxr l a i> p 1 = Q . . . (19) 



Suppose A 2 ±0, /i 3 ±0, /^ = 0, which gives the value of g, then ^c^iPi = ® ^ t7 i°"2i°i = , 

 therefore /) 1 is parallel to a- 1 and p = Ae at o- 1 is a solution of Qp = , A being an arbitrary 

 TRANS. ROY. SOC. EDIN., VOL. XL. PART IV. (NO. 29). 5o 



