She Died A Lady - John Dickson Carr

Rita Wainwright's love affair with Barry Sullivan is flamboyant enough to deserve a dramatic ending, so that when the pair of them vanish over a cliff one rainy night, leaving a farewell note for Rita's husband, no one doubts for a moment that it is a case of suicide - except for Doctor Luke, one of the few people who genuinely liked her.
Sir Henry Merrivale - the fabulous 'H.M.' - is staying in the area, having his portrait painted as a Roman senator. Although confined to a bath-chair with an injured toe, this does not stop him getting about - occasionally in toga and laurels - and solving what is too much for the sharp-eyed doctor.