Mr Pocklington, On the Symbolic Integration, etc. 59 



On the Symbolic Integration of certain Differential Equations 

 in Quaternions. By H. C. Pocklington, M.A., Fellow of St 

 John's College. 



[Received January 2, 1899. Read January 23, 1899.] 



In this paper are discussed (§1) some differential equations in 

 Quaternions of a simple type, but involving V, which can be im- 

 mediately integrated by the artifice of treating V in all respects 

 as a vector. Then (§ 2) some cases are considered in which this 

 method yields a false result. This leads (§ 3) to the considera- 

 tion of the general differential equation </xr = 0, which is analogous 

 to the differential equation with constant coefficients of the 

 ordinary calculus. As an example, a vector is found such that it 

 is equal to its curl. The electromagnetic field of a vibrator in 

 an anisotropic medium (uniaxial) is then (§ 4) worked out in full. 

 The case when the axis of the vibration is perpendicular to that 

 of the medium is especially interesting on account of the unusual 

 form of the functions that enter into the solution. Finally (§ 5) 

 the case when the constant term in the cubic in <£ vanishes is 

 considered. 



1. Consider the equation SVa = 0, where cr is a vector 

 function of the vector-coordinate p of any point. 



If we replace the operator V, which is a vector the constituents 

 of which are differential operators, by a vector 8, the equation 

 becomes SSa = 0, the solution of which is a = V8\, where \ is 

 any vector. If we now replace S by V we get cr = VV\, which 

 should therefore be a solution of the original equation. On per- 

 forming the operations indicated therein we find immediately 

 that cr = W\ is a solution ; and it is clear that it possesses 

 the requisite degree of generality to enable it to be a completely 

 general solution. 



The equation V Vo- = can be solved in the same manner. 

 Since the solution of VScr = is a = 8P where P is any scalar, 

 we may expect that a = VP will be a perfectly general solution of 

 VVa = 0. Here again it is easy to verify this conclusion. 



If S . aVa — 0, a being a constant vector, we can find two 

 forms of solution. Firstly we have cr ± VaV, and therefore 

 a = V ( VaV . X). Secondly, cr, a, V are coplanar, and hence 

 cr = Pa + VQ, P and Q being any scalars. It is easy to verify 

 these results, and to show that they are equivalent to each other. 



The same method can be applied to simultaneous equations. 

 Consider, for example, two of the equations that give the magnetic 



