I read that the hectocotylus arm of an octopus is usually the third
right arm. One of the resources I found counts the arms starting from
the ones nearest to the eyes
(http://www.fao.org/docrep/t0726e/t0726e2m.gif). However I have an
octopus (most likely not a common octopus) specimen in front of me
where the slightly different arm is the second one down. I also saw a
fuzzy picture where the male octopus seems to be reaching out to the
female octopus using that arm. What is the proper convention for
counting arms and which arm is the hectocotylus arm?

Howdy janet05-ga,

A great resource for information on octopuses is the TONMO.com
website.  TONMO stands for The Octopus News Magazine Online.
http://www.tonmo.com/

"Your Octopus, Squid and Cephalopod Information Center"

This message thread on TONMO has more on the usual placement of the
hectocotylus.
http://www.tonmo.com/forums/showthread.php?t=4803

"In this family [Octopodidae] the distalmost portion of the male's
third left OR right (but never both) arm is modified into a thumb
and shovel-like structure, the calamus and ligula, collectively
referred to as the hectocotylus. In some shallow-water species ...
the male also has a thickened portion of web running the length of
either the third right OR left arm (but again, never both) called
the spermatophoral groove."

As for the common convention for numbering arms, etc. there is this
message thread on TONMO.
http://www.tonmo.com/forums/showthread.php?t=1670

"The arms are numbered in pairs (1st left arm, 1st right arm, etc).
Pair 1 is the pair of arms that is most dorsally located, i.e. the
furthest away from the funnel (the 'front' pair in an octopus). The
ventral pair, nearest the funnel, is pair 4 (at the 'back')."

The above points to this TONMO page.
http://www.tonmo.com/science/public/measurements.php

"A Guide to Frequently Used Characters, Character States and
Measures" By Dr. Steve O'Shea.

Figure 2 on the above page shows the placement/numbering of the arms.
http://www.tonmo.com/images/content/measure-fig2.jpg

Another drawing can be found on this TONMO message thread.
http://www.tonmo.com/forums/showthread.php?t=2033

"The hectocotylus can be hard to identify, but I've attached some
sketches at least to help you look for it. For almost all octopuses,
the hectocotylus will be the third right arm of males (it's on the
left in a rare few)."

I could not find any mention of a species of octopus that has their
hectocotylus located on the second arm.  I would strongly suggest
that you post any images of your specimen on the TONMO forum area.
http://www.tonmo.com/forums/

They seem to be a very active and knowledgeable group, and would no
doubt be interested in your specimen and observations.

If you need any clarification, please feel free to ask.


Search strategy:

Google search on: hectocotylus octopus "right arm"
://www.google.com/search?q=hectocotylus+octopus+%22right+arm%22

Looking Forward, denco-ga - Google Answers Researcher