04 
486] 
1 
486. 
NOTE ON DR GLAISHER’S PAPER ON A THEOREM IN DEFINITE 
INTEGRATION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. x. (1870), 
pp. 355, 356.] 
It is worth noticing how easily the case when ф = 1 may be proved independently 
of the general formula with ® ; for (1) the equation 
v = aæ — 
CLi CLo 
IS 
oo oo A»2 oo A^ 
(ax — v) (x—XQ (x — X 2 ) ... — а г (х— \ 2 ) =0, 
and has n + 1 roots, say x 1} x 2 ...x n+1 where 
and (2) the equation 
Xi + x 2 ... + х п л-1 — Xx + X 2 ... + h, n q— 
a 
v = 
a, a 0 
is 
x A*x x ~~ ~X 2 x 
v(x—\ 1 )(x — \ 2 )... — a 1 (x — \ 2 )... — ...=0, 
and has n roots x 1} x 2 ... x n where 
х г + x 2 ... + x n — \x + X 2 ... \ n — 
wherefore 
and 
(Zx + a 2 ... 4- a n 
v 
fv dx x + fv dx 2 ... =fv (dx x + dx 2 ...) 
dv 
=fv 
in the first case 
/. (ax + a 2 ... + a n ) dv . 
=jv - — in the second 
v* 
[which are the two formulae in question]. 
C. VIII. 
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