474 GENERAIi CASE OF THE 



,, ,^, e 1.3, „ 1.3.5.7,4 1.3.5.7.9.11,6 - > 



=^*-fl + ^^'+.a,^/c'+^3-^^^ + , ^c.^, (9) 



jSj, ^3, representing the second, third, ^c. terms. 



Let us resume the differential equation (7) and put s = 



1 11^ 



— — . From this we get 

 i+ku 



putting for the moment 5^ = 4 — 4s + ks\ and thence 

 ^{{i—u').(ii+ku)} = si{i+ku) = ^{s{i—^± -5f5)^,and 



dM= — ;^5l \- —Si ?. Substituting these expressions, 



we shall have 



—6s 



dt=Ki . 



vfs(4— 4s+fcV)|' 



Let s — ms and 2 — ns be the factors of the quadratic 

 4— 4s+)tY in this expression. We shall then have m+n=2, 

 m7i=k-, and thence /«= 1+^(1—*^'), 7i=i—<,/{i—k^). The 

 preceding expression may now be written 



v^«(3— ms).(3 — ns)^ 



which will take the form of the original equation (7) by as- 



suming s= ; since then — ds= — , ^/^ s(2—ms).{2—ns)l=: 



° m m '• ^ 



