THE APPLICATIONS OF QUATERNIONS IN DIFFERENTIAL EQUATIONS. 721 

 we have 



p = Clcr & ■■= b 



if = bp + bp = bb~ l & + ba<r 



= {G(|^70^ 1 F( - ffadt) + G( f^H^f^i - f<t> 1 dt)}& 



-G(|<m^,F(-/<M0°- • ■ • ( 25 ) 



If a solution of either of these equations can be found, the solution of the other is 

 readily obtainable. In the same way in ordinary differential equations we can readily 

 derive the solution of 



aJ-(P + Q/Q)s& + Q;e = 

 from the solution of 



x + Psfi + Qx = . 



22. The methods of solution by infinite series and by definite integrals, which are 

 used in the theory of ordinary differential equations, are applicable, with a few changes, 

 to the theory of vector equations. In this way we can, as in section 4, sometimes make 

 the solution of an ordinary differential equation depend on the solution of a vector 

 equation of lower rank. 



TKANS. EOY. SOC. EDIN., VOL. XL. PART IV. (NO. 29). 6 p 



