APPLICABLE TO ELLIPTIC AND ULTEA-ELLIPTIC EUNCTIONS. 
22-0 
radical of the form — this ratio may always be brought within the limits of 
12 .. 1 
^ and Y ; for, if c sin (p should exceed we may put the radical under one of the forms 
cos<p V^l + (1 — tan^^ or i+rz tan^ (p, and one or other of these new 
radicals will always be less than\/2. I should have remarked that c never exceeds 
unity. 
We may take u as a geometrical mean between v and i;(l — c^sin^^), where 
1 + Nsin^<p c® + N 1 
^ (1 +nsin^<p)(l— c^sin^(p) c^-\-n 1 — c^sin^ip 
n — N 1 
c^-\-n 1 + n sin"^ (p 
We can put x=\ and y = \ — c^sin^(p, or vice versa, as may be convenient. Making 
either substitution, and reducing to partial fractions the mean which we select for inte- 
gration, we have to multiply each fi’action by the second value of v, which doubles the 
number of fractions, and the index of their denominators ; but on again decomposing 
the fractions, and grouping them by their denominators, we reduce them to two more 
than those into which we had previously decomposed the mean. The two extra frac- 
tions will, of course, have \-\-n sin^<p and 1 — c^sin^ip for their denominators. If we stop 
at the thu'd arithmetic mean we shall thus have five partial fractions to integrate, and 
for the third harmonic mean, six. I do not actually exhibit the work or its results, 
because my doing so would not save labour to any one. Not only would the resulting for- 
mula be complicated with constants more easily managed in an arithmetical form, but it 
will seldom happen in practice that it will be worth while to reduce the elliptic function 
to the normal form given above. I have, however, done enough to shoAV that my method 
is capable of approximately reducing any form, containing a function under the radical 
of the square root, to a small series of terms involving, at highest, logarithms or inverse 
tangents in their integrals. Moreover the approximation is so rapid, that, in the case 
of an elliptic integral of the third kind and of logarithmic form, nothing would be 
gained by having recourse to the interpolation of the only possible table, that of the 
double integral 1 . „ . == f v/l — The third pair of means will give 
six or more places of figures correct, and the fourth arithmetic mean is capable of giving 
twelve places. 
Mnth respect to the actual integration of the partial fractions ultimately obtained, 
there is no difficulty. It will easily be seen that each partial fraction will be of the 
form Y ^^” .^ 2 ^ . The integral of this with regard to (p is 
^^7=tan-‘{ v/I+p.tan(p}. 
If \-\-j) is negative suppose), this integral takes the form 
A- loci 
2q^ [1 — 5 tan (pj 
2 H 
MDCCCLX. 
