THE APPLICATIONS OF QUATERNIONS IN DIFFERENTIAL EQUATIONS. 713 



6. The following are some properties of 



F(<£) = 1+ fd<f> + fd<t>fd<f> . . . 

 and 



G(<£) = 1+ ld<t>+ j (d<t>d<f> . . . 



Taking conjugates in (10) we get 



which is of the same form as (9), hence the conjugate of¥(cp) is G((p') . 

 It has been already shown that the solution of 



is 



x =F(-j<t>dtp 



put 



the solution of which is 



= x ' then x 



6=fG(Udt) 



zs and \\i being arbitrary constant linear vector functions. It follows immediately that 



{F(*)}-> = G(-*). 

 Similarly 



(G(l)}- 1 = F(-*). 

 This enables us to put the particular solution of the linear equation of the first rank in 

 the simpler form 



F( - Udt) JG( Udt) x dt . 

 If x = F(<£) we have 



/f -*-*-'«. 



Put 1 — C7 for x, then 



F- 1 (l-^)= -- ffto(l+CT + ^ 2 + CT 3 . . . ) 



the series being convergent so long as the roots of xs are less than 1. A similar result 

 is got on substituting 1 +ct for ^ . # 



* Note on the form of etf>. — If the elements of <f> are ( * 2 3 ) we have 



<f> n = (g 1 n (r 1 S . (r. 2 <r a +g 2 <r 2 S . c^ + ^S . <r l (r 2 )l^xr l a 2< j 3 

 .'. e<t> = (^'o-jS . <r 2 <r 3 + e 9 *<r 2 S . ff 3 (T 1 + ef> 3 a 3 S . < r 1 <r 2 )jS(T l <r 2 (r 3 



and in general /(0) = ^(g^^il^. 



va x o 2 <r 3 



f can also be expanded in the form A<p 2 + B<p + G where A B and C are given by the equations 



Ag* + Bg 1 + C =f( 9l ) Ajr/ + Bg 2 + C =f(g 2 ) kg* + Bg 3 + =f(g 3 ) . 



— (See Tait, Quaternions, 3rd ed., p. 124.) 



