MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
211 
The sum of n integrals becomes 
f x d x V v + J* y dy V v -j- &c. == J ° v' v . S x dx. 
If n is a number greater than 4, S x 2 = 0 : 
.'. Sxdx = 0 .•. sum of n integrals = const. 
If n — 4, S x 2 = 2 v, Sx dx = dv, 
sum of four integrals = J V v . dv = +C. 
If n = 3, Sx 2 = t; 2 , ,\Sx dx = v dv, 
.*. sum of three integrals =y d v = -j v 7 + C. 
But when n is greater than 4, the equation has impossible roots, therefore the solu- 
tion is imaginary. Although, as Legendre has demonstrated*, these imaginary cases 
do not cease to have a real analytical meaning ; the sum of two imaginary integrals 
forming a real integral in a manner analogous to that in which two imaginary roots 
of an equation form a real sum and product. 
But we may avoid these imaginary solutions by putting^/* dx^/i x n [ u the form 
C dx(a + bx+ cx 2 ) yV 1 + ,v ” -. 
V (« + fi,?' + cr y 
Assuming, then, 
] + x n 
(a + b cx 2 ) 2 1 ’ 
we may attribute to the polynomial any number of terms suitable to the exponent ndf, 
and then it is in most cases possible to find such numerical values for the constant 
coefficients a, b, c, See., that the resulting equation shall have all its roots real. 
Each integral, then, has the form 
x {a + b x + c x 2 )\/v, 
and the sum of all 
= / \/ v b x + c x 2 ), 
■where Sdx (a + bx-\-cx 2 ) = the aggregate of the partial sums 
aSd x + b S x dx + c S x 2 dx + ...., 
which is the differential of 
+ + , 
and may therefore be expressed in terms of v, since the quantities S x, S x 2 , S x H , Sec. 
are readily found in terms of v by the usual doctrine of algebraic equations. 
* Fonctions Elliptiques, vol. iii. p. 326. 
71 
f In general the number of its terms may be — or 
w+ 1 
2 
2 e 2 
