117] 
93 
117. 
NOTE ON THE INTEGRAL \dx + J(m — x) (x + a) (# +fr) (a? + c). 
[From the Philosophical Magazine, vol. vi. (1853), pp. 103—105.] 
If in the formulae of my “ Note on the Porism of the in-and-circumscribed Polygon,” 
[115], it is assumed that 
1 
U = x 2 + y 2 + z 2 q— (ax 1 + by' 1 + cz 2 ) 
V = ax 2 + by 1 + cz 1 , 
and if a new parameter &> connected with the parameter w by the equation 
(om 
w — 
m — &) 
be made use of instead of w, then 
w U + V = ——— (x 2 + y 2 + z 2 ) + ax 2 + by- + cz 2 }; 
and thus the equation wU + V = 0, viz. the equation 
o> (x 2 + y‘ 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0, 
is precisely of the same form as that considered in my Note on the Geometrical 
Representation of the Integral jdx+ V(« + a) (* + &)(* + c),’’ [113.] Moreover, introducing 
instead of £ a quantity y, such that 
1 = 
_ 
my 
m — y’ 
dmdy 
v/q! d(m-y)(a + y)(b + y)(c + y) 
then