117] NOTE ON THE INTEGRAL jdx + 7(m-®) (x +o) (x+ 6) (x +c). 95 
which is to be replaced by 
(a+riia + lc) (a -H?) 
~ : — (A/ + lid), 
a + m v rw ’ 
(A±m±m±fi^ + ^ 
These equations give, omitting the common factor (a + m) (b + to) (c + to), 
X, 2 = w? (abc +pkd) 
+ vi{ — a&c (p + k + d) 4- pkd (a+ b + c)} 
+ {abc {kd + 0p + kp) +pk6 (be + ca + ab)}, 
2X/Lt = w? { — (bc-\-ca + ab) + (kd + dp +pk)\ 
+ m { — abc — pled + (be + ca + ab) (p + k+ d) + (kd + dp + pk) (a+ b + c)} 
+ {abc(p +k + d)-pkd(a + b + c)}, 
/x- = m 2 [a + b + c + p + k + d] 
+ vi {(be + ca + ab) — (kd + dp +pk)} 
+ abc + pkd; 
and substituting in 4\ 2 . ¡i- — (2\p) 2 = 0, we have the relation required. To verify that 
the equation so obtained is in fact the algebraical equivalent of the transcendental 
equation, it is only necessary to remark, that the values of X 2 , p 2 are unaltered, and 
that of \p only changes its sign when a, b, c, vi and p, k, d, — m are interchanged ; 
and so this change will not affect the equation obtained by substituting in the equation 
4X 2 . p 2 - (2Xp) 2 = 0. Hence precisely the same equation would be obtained by eliminating 
L, M from 
(k + a) (k + b) (k +c) = (L + Mk) 2 (m — k), 
(p + a) (p + b) (p + c) = (L + Mp) 2 (m - p), 
(d + a) (d +b)(d+c) = (L + Md) 2 (d -p)-, 
or, putting (L + Mk) (m — k) = a + (3k + yk 2 , by eliminating a, /3, y from 
(to — k) (k + a) (k + b)(k + c) — (a + (3k + yk 2 ) 2 , 
(to-p)(p + a)(p + b) (p + c) = (a + /3p + yf)\ 
(to — 0) (6 + a) (d +b)(d +c) = (a + (3d + yd 2 ) 2 , 
0 = (a + fim + y in 2 ) 2 , 
which by Abels theorem show that p, k, d are connected by the transcendental equation 
above mentioned. 
2 Stone Buildings, Jidy 9, 1853.