AND DEFORMATION OF A THIN ELASTIC SHELL. 533 



P. (jt) IB a solution of the differential equation of zonal harmonics 



a ( a + l )P = 0,,V-W^ . (71) 

 and tliis is satisfied by the integral 



This form would not be adapted for arithmetical work if a were imaginary. 



If a be imaginary, then will (+ 1) (i + <f)t where q is real. If q be 

 integral, (71) is the equation of MEHLER'S ' Kegelfunctionen ' ; and it is shown by 

 NEUMANN* that this equation is satisfied by the integral 



i: 



cos </< d<f> 



o ./(/* + cosh <f>) ' 



and this is finite when /t = 1, but infinite when /*= !; the form of demonstration 

 adopted holds equally when q is not integral. 



In general, writing o = a (a + 1), and changing the independent variable to 

 2 = (l /*)/2, the equation for P becomes 



^, 1-2* df P _ 



rfz ^(l -Z) <fe "~Z(1 -2) 



so that one solution for P is the hypergeometric series F(a', ft', y, z) where oi -\-ft =1, 

 a'/J' = a>, y =. 1 ; and this is finite for 2 = or /* = 1, so that 



/l- M \ a 

 -- 



which converges for all real values of p. between + 1 and 1, but diverges for 

 M =-l. 



In our equations the quantity ft is always real ; the quantity a may be complex of 

 the form + iq ; in any case we have always a solution of our equations in series or 

 definite integrals. 



18. Supposing Ti'^/x), Tj, f) (/i) known, we shall be able to write down the values of 

 <r u o-j, OT ; and then, supposing the surface bounded by a small circle /* = const., we 

 have for the boundary-conditions 



"Ueber die Mehler'schen Kogclfunctioncn," ' Mathcmat. Annden,' vol. 18, 1881. 



