from the Picture "The Dolly Sisters"
Music by Harry Carroll
Lyric by Joseph McCarthy
1918
Mr Todd, for his part, was a new man. He had retired to his corner at the end of the preceding round with the moody step of one who sees failure looming ahead. 'I'm always chasing rainbows', Mr Todd's eye had seem to say as it rested gloomily on the resined floor.
Ukridge. 5 The Return of Battling Billson
Or it might be, of course, that a fellow in Brinkley's sozzled condition did not require a definite object of the chase, but simply chased rainbows, so to speak, for the sake of the exercise.
Thank You, Jeeves, 16 Trouble at the Dower House
This reference comes from the Annotations compiled in Madame Eulalie's site