First Virginia Banks Inc. of Falls Church reported third-quarter net income of $10.2 million (69 cents a share), up 15 percent from $8.9 million (60 cents) for the same period last year.

For the first nine months, profits reached $28.7 million ($1.93), up 7 percent from $26.8 million ($1.83) in the 1984 period.

Total assets on Sept. 30 were $3 billion.

*Dominion Resources Inc. reported earnings of $316 million ($3.60) for the year ended Sept. 30, up 6 percent from $297.6 million ($3.55) the year before.

Operating revenue increased to $2.7 billion from $2.6 billion.

Fourth-quarter net income increased 22 percent to $102.9 million ($1.15) from $84.6 million ($1.00). Operating revenue for the quarter increased to $720.1 million from $679.5 million.

Dominion Resources, of Richmond, is the holding company for Virginia Power.

*American Management Systems Inc. of Arlington reported third-quarter net income of $1 million (41 cents), up 23 percent from $812,000 (32 cents) in the 1984 quarter. Revenue increased to $21.8 million from $19.9 million.

For the nine months ended Sept. 30, earnings were $2 million (91 cents), up 5 percent from $1.9 million (76 cents). Revenue increased to $63.8 million from $54.8 million.

AMS applies designs computer systems for corporations, academic institutions and government agencies.

*Jefferson Bankshares Inc. of Charlottesville earned $2.8 million ($1.01) in the third quarter, up 11 percent from $2.5 million (92 cents) last year.

For the first nine months, net income reached $8.2 million ($2.94), up 12 percent from $7.3 million ($2.63).

The company, which operates Jefferson National Bank and four related subsidiaries, had total assets of $1.09 billion at Sept. 30.

*Security National Corp., the holding company of Security National Bank of Washington, reported third-quarter net income of $945,000 ($1.03), up 11 percent from $853,000 (93 cents) in last year's period.

For the first nine months, earnings increased 4 percent to $2.6 million ($2.84) from $2.5 million ($2.73).

*Avemco Corp. of Frederick, Md., reported profits of $1.9 million (62 cents) for the quarter ended Sept. 30, up 90 percent over $1 million (36 cents) in the 1984 period. Revenue jumped to $15.7 million from $12.3 million over the same period.

For the first nine months, net income increased 29 percent to $5.3 million ($1.74), from $4.1 million ($1.37). Revenue increased to $42.8 million from $34.8 million.

Avemco is the parent company of the Avemco Group, which provides insurance, financial, marketing and loss-management services.