this is at once found to be infinite, for finite values of z, only for (z0, s0), its infinite part being log (z − z0), and for z = ∞, for one sign of s only when these are separable, its infinite part being −log t, that is −log z when a0 ≠ 0, and −log (z1/2) when a0 = 0. And, if ƒ(θ) = 0, the integral
| P1 = ∫ ( | s | + z √a0 ) | dz |
| z − θ | 2s |
is infinite at z = θ, s = 0 with an infinite part log t, that is log (z − θ)1/2, is not infinite for any other finite value of z, and is infinite like P for z = ∞. An integral possessing such logarithmic infinities is said to be of the third kind.
Hence it appears that any elliptic integral, by subtraction from it of an appropriate sum formed with constant multiples of the integral J3 and the rational functions of the form (s d/dz)m−1 J1 with constant multiples of integrals such as P or P1, with constant multiples of the integral u = ∫s−1dz, and with rational functions, can be reduced to an integral H becoming infinite only for z = ∞, for one sign of s only when these are separable, its infinite part being of the form A log t, that is, A log z or A log (z1/2). Such an integral H = ∫R(z, s)dz does not exist, however, as we at once find by writing R(z, s) = P(z) + sQ(z), where P(z), Q(z) are rational functions of z, and examining the forms possible for these in order that the integral may have only the specified infinity. An analogous theorem holds for rational functions of z and s; there exists no rational function which is finite for finite values of z and is infinite only for z = ∞ for one sign of s and to the first order only; but there exists a rational function infinite in all to the first order for each of two or more pairs (z, s), however they may be situated, or infinite to the second order for an arbitrary pair (z, s); and any rational function may be formed by a sum of constant multiples of functions such as
| s + s0 | + z √a0 or | s | + z √a0 |
| z − z0 | z − θ |
and their differential coefficients.
The consideration of elliptic integrals is therefore reducible to that of the three
| u = ∫ | dz | , J = ∫ ( | a0z2 + 2a1z | + z √a0 ) dz, P = ∫ ( | s + s0 | + z √a0 ) | dz |
| s | s | z − z0 | 2s |
respectively of the first, second and third kind. Now the equation s2 = a0z4 + ... = a0 (z − θ) (z − φ) (z − ψ) (z − χ), by putting
y = 2s (z − θ)−2 [a0 (θ − φ) (θ − ψ) (θ − χ) ]−1/2