104 Note on the Integral/ fa+ -/(m— x)(x + a)(x+b)(x+c). 



braical equation corresponding to the transcendental equation 

 + IT^ + TifP + 11,0=0. Consider the point f , rj, f on the conic 

 w(# 8 + y 2 +£ 2 ) + aff 2 + 6y 2 .f c£ 2 =0, the equation of the tangent 

 at this point is 



(m -f «)? x + {m + b)rjy + (m + c) & = 0. 



And if be the other parameter of this line, then the line touches 



0(a 2 + y 2 + 2 2 )+a# 2 + ty 2 + <* 2 =O; 

 or we have 



{m + af? , (m + fl)V , (m + c)«(? _ n 



(9 +a + gnp + 6 + c " U; 



and combining this with 



{m + a)P + (m + b)fj*+(m + c)?=0, 

 we have 



f : t; : f= */£ — c \Z« + \/b + m s/c + 



m 



' v(c—a) \/b + 6 */c + m*/a + m 



• V(a — 6) -v/c + V'a + m V^ + ra 



for the coordinates of the point P. Substituting these for x> y, z 

 in the equation of the line PP' (the parameters of which are p, k), 

 viz. in 



x \/b—c \/(a + k)(a+p)+y *Jc—a </{b + k)(b+p) 



+ z s/a — b \Zc+k\/c+p=0, 

 we have 



p. r) S(a+p)(a + k)(a + ff) ] (f , ff) */ (ft +/>) (S + *) (5 + <7) 

 a/« + wi Vb + m 



+ {a _ b) s/{c+p){c + k){c + 6) =0j 

 Vc+m 

 which is to be replaced by 



( a+j ,)( g +*)( a +fl) 



( C +^)( C + A)(c+g) = 



c-fm v T ' 



These equations give, omitting the common factor 

 (a + m)(6 + 7w)(c + m), 



