certain Differential Equations in Quaternions. 65 



the resulting scalar equation, which may be written 



Uff b ('y c (f\*\ 



*~ . »-&+(#+&)•-<-* 



seems to be unintegrable. 



5. Returning to the general case, <p<r = 0, if m 3 = the method 

 of solution given above fails. 



Let the cubic in <f> be 



4> 3 - m$> + m 2 (f> = (8), 



and consider the equation ((f>—g)<r = Q. 



We can express in powers of <£ by dividing (8) by <f>—g, 



and obtain 

 i 

 



4>-9 



g (m 2 -m,g + g 2 ) ' 



In general, on passing to the limit g = 0, the last factor becomes 

 indeterminate, and 



a = (cf> 2 — ?n 1 <£ + m 2 ) \, 



where \ is any vector, a solution containing no inverse operators. 

 If, however, (<£ 2 — m x <£ + ra 2 ) X = 0, we may cancel a g, and then 



a = (cf> — wij) \, 



where (<j> 2 — rrh<f> + m 2 ) \=0, m 2 \ = 0. 



For example, in the case of the equation V.aWa = 0, the first 

 half of the process gives <r = VP, and the second half gives 



<r= Fa/3 ./(/>-«#£), 



so that the completely general solution is 



The whole of the work in this paper can be directly translated 

 into the language of vector analysis, and the examples considered 

 can of course be solved, though with much loss of compactness, by 

 Cartesian methods. 



VOL. X. PT. II. 5 



