182 PROFESSOR A. E. H. LOVE ON THE 



where C is constant of integration. Then we have 



and therefore 



, A r * {A 2 -f(n+6)sy 



l 1.2.(2n + 3) 2 . 4 (2n + 3) (2ft + 5) 



where 0' is a constant of integration. Since -"E n is finite at r = 0, the constant C' 

 must be zero ; but the constant C is in our power, and we may choose it in any way 

 that is convenient. The term contributed to E by C is (2n+l)~ 1 C<u B , which satisfies 

 LAPLACE'S equation, and therefore any change in the chosen value of C is equivalent 

 to borrowing a term of F n to make up a term of <. 

 Now the series 



_ + ._ 



2.(2n+l) 2.4(2 + l)(2n+3) 



satisfies the equation 



and therefore, if we take for C the value 



P - 2n+l . 



fc a +(w+4)s 2 



the function n satisfies the equation 



We shall choose this value for C, and thus we shall have 

 2n+1 



. i^ 4 



2.4(2n+3) 



. A f ___ 1_ ^ {tf+fri+G)*}!* 



L ^+^+4)^ 2(2n + 3) 2.4.(2n + 3)(2n+5) 



(2n + 3) 2.4.(2n + 3)(2n+5) 



/ v 



/ \,+i \i r \n f \rt, ofs |...^< i- {/t,-r6K-r6j a j Q, /^T^ 



2.4...2/c(2n+3)(2n-l-5)...(2n + 2ff-l) 

 and 



2. 4...2K(2n + 3)(2n+5)...| 

 The function E n satisfies the equation 



I dr 



