226 
ME. C. W. IklEEEIFIELD ON A NEW :METH0D OF AFPEOXIMATION 
It is obvious that we may approximate in the same way to the values of ultra-elliptic 
integrals ; but the process will be more lengthy, on account of the greater complexity 
of these functions. 
In the case of higher radicals than the square root, so far as concerns a fii'st approxi- 
mation only, it is clear that, if we insert several means between two integrable functions, 
any given geometric mean will be intermediate in value to the corresponding arithmetic 
and harmonic means ; but, inasmuch as the process aifords no indication of what the 
second step is to be, it does not seem to have any useful apphcation to such functions. 
But it brings all elliptic and ultra-elliptic functions within practical reach of the 
numerical computer. 
ADDENDUM. 
Eeceived March 29, — Eead May 21-, 1860. 
I have thought it advisable, upon reconsideration, to give the approximate formula 
F® 1 
C I 
® 1 + N sin® 
^ .+nsin®<p v^l — c®sm®<p 
derived from the third mean of the arithmetic series : — 
, 1 N + c® 1 
1 1 1 ^ 
1 re — N 
+ ' 4 2-^2^, 
4 4 J 
l-v^l Are 
^ .1 N+ic® /, 1 A 
1 tail" 
’ tan <p 
N + 
1 4 
\/2 
~^4 2— s/2 
H — < 
l-g -y . V ) " tan-'Hl- 
2 - s /'2 
tan ^ 
I also observe that com. log 
2 + n/ 2 _Q.Q.q 
9-93123 06918 42 
log 
2-v/2 _o. 
9-16567 93211 66. 
The application of this formula, in the shape given above, recpiires that 1 + ?? be 
positive, and that c sin (p shall not exceed sin 45°. 
