THE APPLICATIONS OF QUATERNIONS IN DIFFERENTIAL EQUATIONS. 711 



or say 



P + <t>P = X a • • • ( 5 ) 



if the axes of (p are constant, we evidently have 



- (<f>dt - Udt C Udt 

 p = e jS + e I e yadt 



the proof being the same as for ordinary equations. 



If the axes of (p are not constant, this solution does not apply, for in general 



de$ 4= d<f>e$ . 

 Consider first the case where the right-hand side of the equation is zero. 

 The series 



p = (l _ Udt+ j <f> UdP- j <f> /"<£ Udt 3 + . . . )/3 



is evidently a solution, fi being an arbitrary constant vector ; for 



dp= — (f>pdt . 



When the right-hand side is not zero an obvious particular integral is 



P =(i- Ldt+ULdt*- ...)f — J__ 



J J J J {\-Udt+\<}>Ud 



so that, putting 



F(<f>) = l+4>+ J4,<f>dt + j <i> U<pdf- + • • • , 



the complete solution of (5) is 



= F ( - j<fxH)fi + F( - f 4>dt)f{¥( - felt)}- \adt ... (6) 



for it contains three independent arbitrary constants. This method of solution is 

 sometimes of use in the theory of ordinary differential equations. For, if we know the 

 quaternion solution of a system and then reduce the system by Cauchy's method to 

 an ordinary differential equation, the solution of this equation may be deduced from 

 the quaternion solution of the system by resolving the latter along an appropriate axis. 

 For instance, putting 



<p= - aiS.j - bjS.i 

 we find that the complete solution of 



x + Vx + Qx = 



Xadt 



dt* . . . ) 



is 



where 



* = A(1,+ fafbdt 2 + jafbfafbdt* . . . ) 

 + B(jadt+ fafbfadfi+ fafbfafbfadt 5 ... ) . . (7) 



- (Pdt l~Pdt 



a = e and b = - Qe . 



