122 Mr MURPHY'S SECOND MEMOIR ON THE 



this quantity is therefore the sum of the series which we proposed to 

 find. 



Now the equation h. 1. {t) = is satisfied by ^ = 1 ; hence h. 1. t is 



i' t'^ 

 of the form t'. Q, {where Q= — (1 + - + — + &;c.)}, and therefore if we 



Q" 



put — ^ = J^, we get the value of L„ to be 



d".{t''t''''F) 

 df 



which was the formula we had required to verify. 



We may also observe that since in the equation L„ = 0, / must have 

 n values lying between and 1, therefore h.l. {t), according to the powers 

 of which L„ is arranged, must have n real negative roots, which we 

 see confirmed by the positive signs of all the terms which compose L,,. 



9. If we form the equation 



u (1 — h h. 1. u) =t, 

 we have by Lagrange's theorem 



. J..UW.N, ^' d{t\iA.tf ^ ¥ d'{t\\.\.tf , 

 « = . + ;i.h.l.(0+^.-^^^— ^ -f-^-^3.-A^^+&c. 



from whence it appears that Li„ is the coefficient of h" in the value 

 of -^. Similarly if in Article (12) we form the equation 



u \\ -h. (1 - u)] =/, 



du 



we have P„ = the coefficient of h" in -rj . 



10. If Q„ i<? the coefficient of h" in -j-, supposing u to he deter- 

 mined by the equation u{l — hU) = t, U bei)ig a function of u which 

 vanishes when u = l, and T the same function oft, then shall 



j,Q„f ^ x.{x-\){x-2) {x-n + 1) , 



j/F'T 1.2.3 n '^ '' 



