It is difficult to see where a division opperation of two variables is
needed in an early sidewinder.

An analog multiplier in such a situation is likely to saturate at full
output. As I understand it the early sidewinders simply followed the
target and didn't solve an system of differential equations that might
lead to a singularity.

Do you have more data on the nature of the divide by zero?

Most of the analog dividers I have seen actually produce the output Xo
= (X1 x X2)/Y. In other words they are intrinsically dividers and
multipliers. There are 'feedback multipliers', (that also do
division), Dynameter multiplier/dividers that use two moving coils in
coonected by shaft in oppostion that are blanced with a feedback
circuit and photocells, electron beam multipliers (use a cathode tube
with two pair of deflection plates and balance a phosphor dot on a
screen by photocell, 'servo mulitpliers'(slow but accurate), 'heat
trasfer multiliers' (good for 10 hertz), Simulataneous Pusle Width and
Pulse Amplitude multipliers, electronic time division multipliers.
(Dozens more in my copy of 'electronic analog computers by Korn and
Korn of 1956).

http://www.sci.fi/~fta/aim9.html
Reticle Seekers - A Brief Tutorial

The reticle seeker is the most common optical system design employed in
conventional heat seeking missiles. Invented by the Germans during the
latter phase of WW2, the reticle seeker provides a means of using a
single detector element to produce an error signal in rectangular
coordinates, with respect to a point target somewhere within the cone
which represents the field of view of the seeker.

The technique is based on the idea of mechanically chopping the light
flux which impinges on a detector, in such a fashion that the
characteristics of the chopped light pulses vary with the position of
the light source in the field of view. Because the detector produces an
electrical signal directly proportional to the impinging light flux,
electronic hardware can be built to extract a positional error signal
in x/y coordinates, suitable for driving a missile autopilot (or other
tracking device).

The simplest strategy for designing a rotating reticle seeker is the
Amplitude Modulation technique, such seekers being commonly referred to
as AM seekers. In an AM seeker of conventional design, the light
collected by a mirror system is focussed to a spot on the detector. In
between the detector and optics lies a whirling disc of optically
suitable (transparent) material, which has translucent and opaque
patterns etched on its surface, to interrupt the flux of infrared
light.

In an AM seeker, one half of the disc is translucent, and the other
half covered by a spoke pattern, radiating from the centre of the disc.
The result of this is pattern is an optical/electrical signal which is
a series of pulses, repeating with every revolution of the reticle. The
timing of these pulses with respect to the rotation of the reticle
produces a phase signal which is proportional to the position in one
axis, while the amplitude (size or strength) of the pulses provides an
error signal proportional to the position in the other axis.

The limitation of the AM seeker lies in the performance of the AM
detection (here x-axis) circuits, as the average signal from the
detector becomes quite weak in one direction thus producing poor
tracking performance in this axis. A scheme to resolve this is what is
termed frequency modulation (FM), whereby the number of spokes varies
with the radial distance from the centre of the reticle. In this
fashion a target closer to the centre of the reticle produces a smaller
number of pulses per revolution than a target closer to the outer edge
of the reticle. As a result the error signal in the radial axis of the
reticle can be resolved by a frequency discrimination circuit which is
locked to a reference frequency signal produced by the reticle motor.

Practical seekers use a range of variations on these two themes, with
various schemes using fixed cassegrainian mirrors and moving reticles,
or rotating secondary mirrors and fixed reticles, the latter
arrangement used in the Sidewinder family.

Other design issues in reticle seekers revolve about the detector
element, its supporting optics and cooling system employed. The
detector is a small piece of semiconductor material with suitable
photo-electric properties, ie it changes its electrical resistance or
produces an electrical current or voltage when illuminated. The key
design parameters in choosing a detector are sensitivity, a measure of
how faint a light signal will generate a useful electrical response,
and colour sensitivity, a measure of which visible or infrared
wavelengths will or will not produce a response. Most semiconductors
used for the purpose have some characteristic longest wavelength to
which they respond, while producing output for all shorter (hotter)
wavelengths. Therefore some detector materials can see only hot objects
like tailpipes, whereas others can see the whole aircraft.

Because all hot objects, such as the sun or flares, emit infrared
blackbody radiation, a missile seeker must have means of reducing or
removing such sources of infrared light to prevent seeker seduction.
Therefore optical filters are used. These filters are typically made of
a rare earth doped glass, with a multiple layer interference filter
deposited on the surface. Such filters are essentially transparent over
a narrow range of colours and opaque to all others, therefore passing
only the desired infrared colour through to the detector.

Cooling the detector is a means of improving its sensitivity. Even the
meagre amount of heat in a detector at room temperature will produce a
response in a good material, resulting in thermal noise which would
mask the target, therefore the detector must be cooled to prevent this.
Two strategies are typically used for this purpose, thermoelectric
cooling with a Peltier device or gas cooling. A Peltier is a
thermocouple which acts as a heat pump, albeit very inefficient, when
electrical current is passed through it. Gas cooling relies on the
expansion of compressed gas, and while lighter than Peltier schemes,
usually imposes a limit on total seeker cooling time when the gas
bottle is exhausted.

The evolution of heatseeking missiles over the last four decades has
seen almost every one of these schemes, or combinations thereof
employed. The Sidewinder is a good instance.