Bessel- Clifford Function, and its applications. 505 

 where I, called its differential invariant, is given by 



T = Q-S- p2 ' < 2 > 



the original differential equation being 



^2 + 2P — ■+ i^u = 0, or R, a function of z ; (3) 



and when I is some power of c, the equation is reduced 

 at once to the Clifford form by a mere change of the 

 independent variable, whereas the dependent variable 

 requires to be changed as well if the reduction is made 

 to Bessel's form. 



Writing the equation, with I — kz m , 



\ % + **" = 0, or z> g + kzPy = 0, /, = m + 2, (4) 



and changing the independent variable to x = kz p /p 2 , the 

 differential equation changes at once to the form in (5) § 1, 

 with n= — 1/p. 



The reduction is equally simple for the more general 

 form 



£(*£) + *■> «0, or ^g + j.J + fa-^-6, (5) 



by changing to the new independent variable 

 x = A2 OT -* +2 /fm-g + 2) 2 , 



.£-<.-9+*>& } (6) 



*»? f + (-^%+l)«? + ^ = ) ... (7) 



aar \>n — (j + 2 J dx ° v ' 



■" = °-<*>' » = ,^T2 < 8 > 



This is the differential equation arising in the stability of 

 a mast or tree, a pile of books, or Eiffel tower ; and for a 

 vertical wire or uniform mast, ^ = 0, m=l, n=—% % as 

 investigated in the Proc. Cambridge Phil. Soc. vol. iv. 1881. 

 The simplification in the results is evident, if the Clifford 

 Function is used, instead of Bessel's form. 



