Southern Railway Co. yesterday reported a modest gain in profits for the second quarter of 1981, despite sluggish business during the 73-day United Mine Workers strike. Separately, UNC Resources reported a small loss in the three months ended June 30, and Financial General Bankshares Inc. said earnings in the period rose 58 percent.

Washington-based Southern listed profits of $52.5 million ($3.36 a share), a record for the quarter and 6 1/2 percent higher than $49.3 million ($3.16) in the same period last year. Operating revenues rose 6.6 percent to $438 million.

Coupled with a record performance in the first three months of the year, Southern's earnings for the six months ended June 30 were up 11 percent to a new high of $111.6 millon ($7.14), compared with $100.6 million ($6.45) as revenues increased 8 percent to $875.6 million.

President Harold Hall said the higher revenues resulted entirely from higher freight rates as coal revenues fell 16 1/2 percent, but revenues from all other major commodities sectors increased. He also noted that coal loadings have rebounded since the strike ended and are now running ahead of last year. If this trend continues, it should help offset a 6 1/2 percent decline in overall freight ton-miles in the first half of 1981.

Directors approved a regular quarterly dividend of $1.06 a share, payable Sept. 15 to owners of record Aug. 14.

UNC Resources of Falls Church, a uranium mining and tool manufacturing firm, reported a loss of $734,000 in the first fiscal quarter ended June 30, compared with a steep loss of $4.1 million a year earlier. Revenues fell to $48.1 million from $75.6 millon because of a sharp decline in uranium deliveries.

President Keith Cunningham said the first-quarter improvement over last year is in line with expectations for a profitable year. Outside of uranium operations, where UNC reduced unit-production costs sharply, other results were good in the recent quarter, Cunningham stated.

Financial General Bankshares of Washington, a bank holding company that controls the First American banks in D.C., Maryland and Virginia, said second-quarter operating profits totaled $5.2 million (72 cents a share), compared with $3.3 million (46 cents) a year earlier. Following similar record earnings in the prior three months, Financial General's six-month operating profits were up 29 percent to a record $10.1 million ($1.39) from $7.9 million ($1.08).

Chairman B. Francis Saul II attributed the gain primarily to a 19 percent growth in interest-transaction profits what reflected the recent record level of interest rates as well as growth in short-term investments by the bank firm. Return on shareholders' equity (stock investment) in the bank rose to 14.1 percent from 12 percent.

Total assets of Financial General banks rose 11 percent in the past 12 months to $2.5 billion on June 30, with deposits up 8 percent to $2.1 billion and loan volume up 2 percent to $1.29 billion.