712 MR J. H. MACLAGAN-WEDDERBURN ON 



This method has some interesting applications, which I propose to develop in a separate 

 paper. In particular, it will be found to lead in an easy and natural manner to 

 Bessei/s and Neumann's functions. 



5. The linear equation of the first rank 



p + <£p = 

 can be put in the interesting form 



K + ,-0 . . . (8, 



where 



if/ = I <pdt 



and v- is defined as (cty) _1 cfy> ; for 



d dxf/ d d 



dt dt dif/ di// ' 



If the axes of \f/- are constant, the solution of (8) is evidently 



P = e-*{3 



where /3 is an arbitrary constant vector. 



In this case ^ may in many respects be treated as a scalar, for d^ and \^ are 

 coaxial, and therefore commutative. 



If the axes of -^ are not constant, the solution can be obtained in the form 



P = (l- Jd\lf+ jdij/jdif/ . . . )/3 

 = F(-^)/3. ' 

 Another interesting case arises if we assume p = x a where a is an arbitrary constant 

 vector, then 



X + 4>x = . . (9) 



the solution of which is 



X = F(- f<pdt)7S 



where zs is an arbitrary constant linear vector function. 

 A connected form is 



x + x<£ = . . . (10) 



the solution of which is 



X = nr(l - (<pdt+ I [<pdt<pdt- J j I <f>dt<pdi<pdt . . . ) 

 = CTG( - Udt) . 

 Another equation of similar form is 



Its solution is easily found to be 



x = ¥(-j<pdt)G(-j<pdt). 



