Reading the text printed on the reverse of the mount provides an explanation of the purpose of the carte:
This celebrated picture was purchased by Messrs. AGNEW, at the sale of the late Mr. WYNN ELLIS'S pictures, for the enormous sum of £10,605. When on exhibition in Bond Street, it was stolen on the night of the 25th of May, being cut out of the frame. £1,000 is offered by Messrs. AGNES for the recovery of the picture, and conviction of the theif or thieves. Photographed by kind permission of Henry Graves, Esq., Pall Mall.Wikipedia, in an article entitled, "Famous Cases of Art Theft," provided an answer as to what eventually happened to it:
In 1878, burglar Adam Worth stole Gainsborough's The Duchess of Devonshire from London art dealers Agnew & Agnew, which he used to negotiate the release of an accomplice from prison. However, as Worth's friend had already been freed, he demanded a ransom instead, which would finally be negotiated for an undisclosed amount in 1901.The portrait, which was eventually purchased in the 1990s by Andrew Cavendish, 11th Duke of Devonshire, for the Chatsworth collection, is partly reproduced in an article about the Duchess's life on Wikipedia. It was once said of the Duchess that, "a man could light his pipe by the fire in her eyes"! A more complete story of the fascinating history of the painting is given in a chapter of Joseph Geringer's book, Adam Worth: The World in His Pocket (The Crime Library).