BDM International Inc., a McLean-based professional services firm that does most of its work for the U.S. military, reported record profits for its second quarter, posting a 31 percent earnings increase for the three months ended June 30.

Second-quarter net income rose to $2.5 million (25 cents a share) from $1.9 million (19 cents) for the 1984 period, the company said. For the first six months, net income totaled $4.6 million (46 cents), a 27 percent increase from $3.7 million (37 cents) in the first half of 1984. Revenue for the first half was up 29 percent, from $87.3 million to $112.8 million.

*The Eastmet Corp. of Baltimore said losses continued into the second quarter of this year, although the company said the rate the red ink has been flowing has declined steadily over the first six months of the year.

Eastmet said it lost nearly $4 million (79 cents a share) for the quarter ended June 30, compared with profits of $1.3 million (25 cents) for the same period last year. Sales during the period dropped from $69 million to $59.4 million.

The company said losses for the first six months totaled $10.5 million ($2.09), compared with a profit of $2.9 million (57 cents) for the year-previous period.

Eastmet President William F. Dausch said the relative improvement over the first quarter was the result of favorable trends at Eastmet's largest operating unit, the Eastern Stainless Steel Co.

*Equitable Bancorp of Baltimore said it earned $8.2 million (83 cents a share) in its second quarter, up from $5.8 million (58 cents) in the 1984 quarter. For the first six months of 1985, earnings were $15.6 million ($1.65), up from $10.8 million ($1.04) for the 1984 half.

The bank holding company had assets of $3.5 billion at the end of last month, a 19 percent increase from the same date last year. Total loans have grown 27 percent from last year's second quarter, to $1.7 billion at June 30. Equitable has 134 offices in Maryland and Delaware.

*VM Software Inc. of Vienna said net income increased slightly in the second quarter to $591,000 (16 cents a share) from $527,000 (15 cents) for the same period last year. Revenue increased to $4.2 million from $2.7 million during the same period. For the first six months of the year, revenue jumped to $7.4 million from $4.4 million, while earnings rose to $909,000 (25 cents) from $684,000 (19 cents).

VM Software manufactures software products for use with the VM operating system, an IBM interactive mainframe operating system. The company made its initial public offering on May 29.

*Realty Income Trust reported net earnings of $157,000 (11 cents a share) on revenue of $1.1 million for its fiscal fourth quarter ended April 30. For the same quarter last year, the company earned $220,000 on revenue of $1.2 million.

For its fiscal year, Realty Income Trust said it earned $1.6 million ($1.05) on revenue of $3.4 million, compared with earnings of $1.5 million (94 cents) on revenue of $3.6 million in fiscal 1984. The Chevy Chase company collects rents and mortgages from property it owns.

*American Filtrona Corp. of Richmond reported net income of $1.6 million (84 cents a share) on revenue of $27.2 million, compared with net income of $8.7 million ($3.96) on sales of $27.5 million for the second quarter last year. The 1984 profits are skewed because they include $7 million in capital gain from stock sold back to Bunzl P.L.C., a London company, said the maker of plastics products and filters.