thoroughbred

Most likely Eleanor M. Cochrane, the wife of Richard Sears. Both are sports people. en.wikipedia.org/wiki/Richard_Sears_(tennis)

An article with photo about her is in The Spokane press., April 08, 1910, Page 3. Because it is reported that among the letters she received were several marriage proposals, I think she was not married. Another photo of her (on a horse) is in the Evening public ledger., September 22, 1920, SPORTS EXTRA, Image 30. In the text she is described as well-known society girl of Boston.

"Eleonora Randolph Sears (September 28, 1881, Boston – March 16, 1968 Palm Beach) was an American tennis player of the interwar period. In addition, she was a champion squash player, and prominent in other sports; she's considered one of the leading all-round women athletes of the first half of the 20th century. ... Sears was the daughter of Boston businessman Frederick Richard Sears, a cousin of Henry Cabot Lodge, and a great-granddaughter of Thomas Jefferson." en.wikipedia.org/wiki/Eleonora_Sears

The same woman:

Other Bain photos of her:

The picture on the right is embossed by the same photographer as this one, Harold A. Taylor, Coronado, and Ms. Sears is wearing the same clothes.
see also
www.nwhm.org/education-resources/biography/biographies/el...
Taylor was a photographer in Coronado, California.
contentdm.califa.org/cdm4/results.php?CISOOP1=all&CIS...
Looks like this may have been taken in Coronado, California on March 3, 1915, during polo matches there. A contemporary account mentions all the women in the group picture:
"In the ladies' game Miss Eleonora R . Sears, of Boston, and Mrs.
C. C. Rumsey, formerly Miss Mary Harriman, daughter of the late
railroad magnate, E . H . Harriman, appeared on one side . On the
other were Miss Kathryn Steele, of New York, sister of Mrs . Devereux
Milburn, and Miss Louise Fleischmann, niece of Colonel Max Fleischmann,
captain of the Cincinnati polo team. Miss Sears ' side won
by a score of 3 goals to 2 goals, the winning goal being registered by
herself in the last few minutes of play.
The teams in the round-robin were nominated by Mrs . Devereux
Milburn, Mrs . J. Watson Webb, Mrs. C . C. Rumsey and Mrs . Max
Fleischmann . Mrs. Rumsey's team won the match, and C . Perry
Beadleston, a member of her team, made the highest individual score
with 6 goals . "
www.hpa-polo.co.uk/yearbooks/1915 Mar - 1915 Aug.pdf
Here are some accounts of Sears at polo pony races at the Coronado Country Club in 1912: "Miss Eleonora Sears, the athletic "bachelor maid" of Boston, rode Joe Ross to victory in a quarter-mile dash on the Coronado track to-day."
query.nytimes.com/gst/abstract.html?res=F00F1FFD3F5813738...
Another account describes how she raced in men's attire while her opponent wore a riding skirt.
cdnc.ucr.edu/cgi-bin/cdnc?a=d&d=SFC19120312.2.112.8#
cdnc.ucr.edu/cgi-bin/cdnc?a=d&d=SFC19120310.2.96
A modern assessment of her life:
www.thedailybeast.com/articles/2012/06/22/the-mother-of-t...
and see

His wikipedia entry: en.wikipedia.org/wiki/Foxhall_P._Keene

"Cornelius Vanderbilt III (September 5, 1873 – March 1, 1942) was a distinguished American military officer, inventor, engineer, and yachtsman, and a member of the prominent American Vanderbilt family." en.wikipedia.org/wiki/Cornelius_Vanderbilt_III

Cornelius Vanderbilt was a member of the New Yoirk National Guard for over thirty years, This photo (or an almost identical one) was used in NYT of Sept 26, 1915 select.nytimes.com/gst/abstract.html?res=F10714F73B581273... under the headline of "Scenes at National Guard Civil War spectacular in Van Cortlandt Park.