In 1863, the favorite hostess of Washington society, Kate Chase, the beautiful daughter of Secretary of the Treasury (and later Supreme Court Chief Justice) Salmon Chase, married William Sprague, a wealthy textile manufacturer who had been wartime governor and then senator from Rhode Island. Sprague had a reputation as a drunkard and a very short-tempered man. Kate may have thought she could reform him--or more likely thought that whatever her new husband's faults, the important thing was that he had money and political influence which could help her father become president of the United States.
In any event, after the death of her father and the collapse of the Sprague fortune in the depression of 1873, Mrs. Sprague became involved in an affair with Senator Roscoe Conkling, the vain, handsome leader of the Stalwart (pro-Grant) faction of the Republican party. (Conkling had long been estranged from his own wife.) Conkling and Mrs. Sprague didn't make much of an attempt to keep the affair secret. They were often seen together, both in Washington and Europe, while Mrs. Conkling attended to her garden in Utica, New York. When Kate's financial problems became serious, Senator Conkling introduced a bill to exempt the Chase estate from taxation--explaining that of course he was motivated by reverence for the accomplishments of the late Chief Justice....
In August 1879, the affair suddenly took a sensational turn, summarized as follows by Thomas Reeves, *Gentleman Boss: The Life of Chester Alan Arthur* (pp. 152-3):
"Conkling was a guest at the Sprague home in Rhode Island during one of the former governor's many absences, when suddenly Sprague appeared at three o'clock in the morning carrying a shotgun. The first version of the story had it that a servant warned Conkling in advance, and he was able to flee through a window, leaving behind his belongings. At any rate, the next morning Conkling was again chased from the house by the armed husband, and later, at a nearby restaurant, there was an ugly scene between the two men that was observed by several fascinated bystanders."
https://books.google.com/books?id=0vKI0honHdoC&pg=PT235
The story of Conkling's escape through a bedroom window clutching his trousers was of course widely circulated by Conkling's many enemies, but as much as it must have embarrassed him, it did not cause him as much political damage as one might expect. Conkling's roles as Senator and New York party boss were after all dependent on the support of politicians, and only very indirectly on popular support. Despite the Sprague incident and the Stalwarts' loss of control of the New York Customhouse, Conkling emerged from the 1879 elections with his political allies controlling the governorship and state legislature. He was not able to get Grant nominated for President in 1880 but at least he was able to stop his arch-enemy James Blaine from getting the nomination, and the Stalwarts were sufficiently important that the eventual nominee, Garfield, felt obliged to offer the vice-presidential nomination to a Stalwart--either Levi Morton (who turned it down on Conkling's advice) or Chester Arthur.
But suppose a drunken William Sprague had indeed killed Conkling. Could the Stalwarts have survived the loss of their leader? In any event, they would be weakened to the point where it seems very unlikely that Garfield would choose Arthur as his running-mate, and I don't think he would choose Morton, either--though if he did, Morton would be more likely to accept without Conkling around to advise him not to. (Assuming that Garfield was elected, and that neither Arthur nor Morton nor any other Stalwart was vice-president, would Guiteau have killed Garfield? True, Guiteau was crazy, but even crazy people often act "logically" within their own frame of reference, and Guiteau did consider himself a Stalwart and did consider shooting Garfield a pro-Stalwart act. Would he have shot Garfield had there been a different vice-president?) But would Garfield have been nominated in the first place? Without Conkling, the Grant forces might have been weaker and therefore unable to stop Blaine.