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XXIX. — On the Applications of Quaternions in the Theory of Differential 



Equations. 



By J. H. Maclagan-Wedderburn. 



(MS. received July 7, 1903. Read July 20, 1903. Issued separately October 30, 1903.) 



The object of this paper is twofold : in the first place, to classify and sytematise 

 vector differential equations ; and in the second place, to show the applicability of 

 quaternions to the theory of differential equations. So far as I am aware, the only paper 

 of importance on the subject is one by Tait. # He, however, deals only with simple 

 cases. 



In the classification of forms I have followed in general the treatment adopted by 

 FoRSYTH.t Reference will also occasionally be made to Jordan, j In what follows the 

 order of the highest occurring differential coefficient will be called the rank of the 

 equation ; while the order of the equation will be used, as usual, to denote the order of 

 the equivalent normal system. 



Differential Equations of the First Rank. 

 1. Variables Separable — (Forsyth, § 13). — The general form is 



\p(p , dp) = <pa . (it . . . ( 1 ) 



where \j/- is any vector function of p and dp, which is linear in dp and (pa is any vector 

 function of t (a vector function of t can always be put in this form, a being any fixed 

 constant vector, and (f> a variable linear vector function). 

 The solution of the equation is obviously 



^(p,dp)=l<lt<pa + fl 



where ft is an arbitrary constant vector, but the integration can only be carried out 

 when ^(pdp) is of the form 



f(p,dp) = Sd P V .(F P ). 

 A particular case is 



xf/{p n ~ x dp) = <padt 



* " Note on Linear Differential Equations," Proc. R.S.E., 1870. Scientific Papers, vol. i. p. 153. 

 t Treatise on Differential Equations, 1888. 

 X Cours dl Analyse, i.-iii., 1887. 



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



