122 Mr. Lubbock on some Elemefitary 



Suppose w = 4, then the equation 



x*^^ — Px+ Q = has only three roots, x^, ar^, x.^ 



x^ -\- x^ -\- x^ =: 



JT| x^ x^ 



Xi X^ •+• -^2 "^S "•" '''I '*^3 ^ '"' T 



111 Xi Xn Xo 



or, — + — + — = \ » 



Xs x^ x^ 4. 



These are Mr. Talbot*s equations of condition in the paper 

 to which I have referred. 



Since e^ x/TTx[* = l-^c^x^ 



c, = Sj V 1 + 0^1 — 1 = -^—^ — 



Xi Xa X^ 



2 " ^1^ 



•X^o ~\ *^o *— ■"" w 



If x^ Xs = 2A 

 x^ and x^ are found by solving the quadratic 



x^ — xxi+2A=zO 



when ^1 is given. 



Making Xi = 2 — = '5 ^i = — 1 



I find a-g = - 2-88721 — = - -346355 ^s = + 1 



x^ 



•88721 — = 1-12713 63=—! 



X 



Making x^ = 5 — = '2 gj = — 1 



I find ^2 = - 5-38645 — = — -185651 63 = + 1 



Xq = -38645 — = - 2-58765 63 = -1 



Xg 



In order to obtain convergent expressions for the integral 

 required, I make 



TT? = «. then 



/> dx i 3 4 , 3.7 „ , 3.7.11 1' , „ 



/TT^* = " + *T5 »+ 47879 "' + 4.-8.V2713" ' +«"=• 



