94 NOTE ON THE INTEGRAL J dx = J (m — x) (x + a) (x + b) (x + c). [117 
Also £ = ao gives rj = m, the integral to be considered is therefore 
ii/77= , - ... ; 
J m v (m — 77) (a + 77) (6 + rj) (c + 77) 
i.e. if in the paper last referred to the parameter 00 had been throughout replaced 
by the parameter to, the integral 
dr) 
V(a -f 77) (b + 77) (c + 77) 
would have had to be replaced by the integral IT / y. It is, I think, worth while to 
reproduce for this more general case a portion of the investigations of the paper in 
question, for the sake of exhibiting the rational and integral form of the algebraical 
equation corresponding to the transcendental equation ± 11/& + + 11/0 = 0. Consider 
the point f, 77, £ on the conic to {x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0, the equation of the 
tangent at this point is 
(to + a) + (to + b) 77y + (m + c) &= 0 ; 
and if 6 be the other parameter of this line, then the line touches 
0 (x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0 ; 
or we have 
('m 4- a) 2 i- 2 (to + b) 2 y 2 (to + c) 2 £ 2 
6 + a + 6 + b 0 + c ’ 
and combining this with 
(to + a) £ 2 + (to + b) 77 2 + (to + c) £ 2 = 0, 
we have 
£ : 77 : £ = V b — c da + d^b+mdc+m 
: V(c — a) Vi + 6 Vc + to Va + to 
: d(a — b)dc + 0da + mdb+m 
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 db — c \!{a + k)(a +p) + y Vc — a V(6 + k) (b +p) + zda — bdc + kdc+p = 0, 
we have 
(b _ c ) (a + ^) (a + 0) + ( c _ ^(b+P)(b + k) (b + 0) 
da + to V6 +m 
+ ( n -6) V(c+J>) , ( lP )(c + g) = o, 
vc + m