MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
15 
On the other hand we have 
2 Q = 2 arc (90°) = 2*5224 
~ e 2 r = 0*3246 
2 Q - e 2 r = 2-8470 
Sum of arcs = 2-8471 
Error = 00001 
I will now indicate two other theorems respecting the sum of three elliptic arcs. 
/ / 1 — OC^" 
d X t ^ in the form 
f(l+ex)dx\J 
and assume 
1 - p pQ t 1 e • 
\ — " x ^\ a symmetrical = — . This gives 
(1 + e x) ( 
2 ) 
& + t * 2 — ( v + *) x + v —r L — °> 
and the result which I find is, that if three abscissae are the roots of this equation, 
the sum of the corresponding arcs = 2 e v + C. 
II. We may put the integral in the form 
/* d x / (1 4 - x) (1 — e* x 2 ) 
J 1 + x v 1 — x 5 
and assume ^ 1 + ^ ^ x e X ^ — v, whence 
+ x 1 — V A i - x + 
0. 
The result which I find is, that if three abscissae are the roots of this equation, the 
sum of the arcs — 2 J v + C. 
These theorems respecting the sums of elliptic arcs appear to be some of the simplest 
which exist ; but an unlimited number of theorems of a higher order and more com- 
plicated nature are obtainable, the discussion of which would lead too far at present. 
Thus if we assume 
/ 1 — e 1 X 2 n- 1 , 
\/ \ # 2 ' — a n -i x + a n _ 2 
x n “ -j- &c. 
where the coefficients are constants, or entire rational functions of v, we have an 
equation of 2 n dimensions, which gives the sum of 2 n elliptic arcs in terms of v. 
There is no difficulty, beyond the length of the operation, in deducing these 
theorems, as they are all obtainable by an uniform method. But it will be of im- 
portance to show the relation between them and the previously received doctrines 
respecting elliptic integrals as established by Legendre and others, the connexion 
between them not being at first sight very evident. 
