The two main bank players in the U.S. on-line brokerage market posted starkly different results during the third quarter, according to separate studies.

Waterhouse Investor Services, the New York-based broker-dealer unit of Canadian Toronto-Dominion Bank, is making strides in Internet trading, while Fleet Financial Group's Quick & Reilly is losing market share, according to studies from Credit Suisse First Boston and Piper Jaffray.

Piper Jaffray, Minneapolis, ranked Waterhouse fourth among on-line brokers nationwide, saying the firm handled 10% of on-line trades. Charles Schwab & Co. was first, with a 29.1% share, followed by Fidelity Brokerage, and E-Trade Group, Piper Jaffray said. Quick & Reilly came in seventh.

Credit Suisse First Boston, meanwhile, put Waterhouse in third position, with 10.5% of the on-line market, behind Schwab and E-Trade but ahead of Fidelity. Quick & Reilly ranked eighth.

At the end of the second quarter, Waterhouse held roughly 9% of the market, according to both Credit Suisse First Boston and Piper Jaffray.

Stephen C. Franco, an analyst with Piper Jaffray, attributes Waterhouse's success to the firm's ability to encourage customers to use the Internet to trade their accounts.

"We've done a good job," said Frank Petrilli, president and chief executive officer of Waterhouse. "We give people the power of choice." Customers can access accounts over the phone or through 150 branches, as well as on-line, he said.

Being named the top broker in Smart Money's annual survey of on-line brokers has also helped, Mr. Petrilli said. Added Mr. Franco: "That's the best word of mouth you can get."

Both reports found that Quick & Reilly's grip on the on-line market has weakened. Credit Suisse First Boston said the firm controlled 3.8% of the market on Sept. 30, versus 4.3% on June 30. Piper Jaffray put Quick & Reilly's share at 4.3% on Sept. 30, down from 5% at the end of June.

"Of all the broker-dealers, they haven't done a good job migrating their customers to on-line trading," Mr. Franco said. He said that Quick & Reilly has diluted its market share by creating SureTrade Inc., a deep discount franchise that caters to frequent traders.

A Quick & Reilly spokesman said that in recent times the firm has spent more on marketing SureTrade, preferring to rely on the brand recognition of Quick & Reilly for its Quickway Net service.

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