I am starting investing in stocks. I know that some companies are
traded on multiple stock exchanges around the world. For example,
Philips Electronics has different tickers in New York and Amsterdam -
and its shares are nominated in USD and EUR respectively. But the
share price is not necessarily proportional to EUR/USD ratio (example:
TGEN, Paris EUR10; TRGNY, NYSE USD4). Why?

Small variations would be due to most daytraders and specialists
working on only one stock exchange. If the differences were often more
than 1%, I should think money could be made by switching exchanges. 
Neil

This may be of interest.
http://www.smartmoney.com/ask/index.cfm?story=20000712

Hi a_n,
In a theoretical world stocks in different stock markets for the same
company should trade at the same price. The process of arbitrage
balances these differences where institutional investors are able to
buy at the stock exchanges where the stock is cheap and sell it at the
stock exchanges where the stock is more expensive.
BUT arbitrage in international markets is not an easy game. Some of
the reasons that stocks cannot be instantly bought and sold to
leverage the differences in prices are:
1) Stocks of foreign companies do not trade in the US directly. Shares
of such companies are owned by Banks which issue American Depositary
Receipts (ADRs). These ADRs trade in the US. SO one cant really buy a
stock on the LSE and instanty sell it on the Nasdaq.
2) Time zone differences allow trading in different stock exchanges at
different times.
3) Fluctuations in foreighn exchange rates add to the risks of
international arbitrage.
4) Often you need to own a stock before selling it. So by the time you
buy a stock and it is transferred into your account the price of the
stock in the other market may have shifted.
To conclude, the price differences remain because there is no "quick"
way of balancing out these differences. And then again as a new
investor you may consider specialising in arbitrage as th erisks come
with their own rewards.