The $100 million public stock offering of the nation's largest railroad. Burlington Northern Inc., was fully subscribed yesterday within hours of its availability.

Sydnicate department brokers at the issue's leading underwriter. Morgan Stanley & Co., as well as at Shearson Hayden Stone Inc., a secondary underwriter, confirmed that the response to the railroad's offering was unexpectedly enthusiastic.

The rapid subscription signalled some restortation of investor confidence in the financially troubled railraod industry, as well as Burlington Northern, Wall Street analysts said yesterday.

Investors purhcased 2 million shares of $2.85 convertible preferred stock at $50 a share. Each preferred share is convertible into 0.8889 shares of the line's common stock. The comversion price would be about $56.25 a share.

The proceeds of the sale are earmarked for the repayment of about $35 million of debt incurred under the railroads's revolving credit agreement, for the financing of capital improvements, the acquisition of rolling stock and the expansion of freight operations especially for coal traffic.

The prospectus on the offering said the line plans to spend $2.2 billion during 1977-8: on capital improvements with about $1 billion of that to prepare for the anticipated increase in coal traffic.

The railroad is the largest in the U.S. in terms of total track miles and second in terms of transportation revenue.

Its lines extend west and north from Chicago to the Pacific Northwest and south through Denver to the Gulf of Mexico at Houston and Galveston.

While its financial performance has been stronger than other carriers in the Seventies, Burlington Northern is a major land owner with significant lumber, mining oil and gas holdings and is not totally dependent on freight and passenger revenues.

Last October the board increased the quarterly dividend to 40 cents a share from 30 cents. It registered 1976 profits of $73.03 million ($5.69) on revenues of $1.9 billion. For the first quarter of 1977, the lines reported a 15 per cent increase in revenues over the year earlier period to $496.02 million with profits of $43.09 million $3.45 a share).

Yesterday's offering apparently was unaffected by a signficant rebuke to Burlington issued by the Securities and Exchange Commission.

Ten weeks ago, the SEC ordered the carrier to make more adequate disclosure of its financial position in its public reports and to modify its accounting pocedures.

The agency said the procedures used by the railroad did not reflect the decline in the profitability of its freight business since 1970, nor the degree of deferred rail facility maintenance.

The SEC decision was the first significant statement about rail corporation issued since the agency began overseeing the industry's fiancies in 1976. Prior to that, the railroads were subject to virtually nonexistent Interstate Commerce Commission oversigh.

Charges that the line made inadequate disclosure were issued in 1974, when multimillionaire industrialist Norton Simon resigned from the Burlington board of directors charging that the railroads management was "screwing the American public."