The October 1786 edition of The Gentleman's Magazine featured an ode written to the memory of "Lady Harriet Elliot" [sic], the recently deceased sister of William Pitt. Her name was actually spelled Lady Harriot Eliot, therefore it is unlikely to have been written by someone who knew her personally. She died of fever at Downing Street on September 25, 1786, just five days after giving birth to her first child. Her daughter was named Harriot Hester. I've previously written about the funeral of Lady Harriot Eliot and an examination of her funeral costs.
In the words of Lord Sydney to his son, Lady Harriot's husband Edward James Eliot was left "almost frantic" by his wife's untimely death . An ode penned to her memory was listed in "Bibliotheca Cornubiensis: Comprising a supplementary catalogue of authors, Lists of Acts of Parliament and Civil war tracts, &c., and an Index to the contents of the 3 vols" (1882), as having been written by Edward James Eliot, albeit under the name of "J.N. Puddicombe, Dulwich College." Although I question whether this poem was actually written by Eliot himself, it is extremely moving, and the sentiments expressed within appear to be genuine. I'd like to extend a hearty thanks to the lovely Sarah for bringing this to my attention.
I've reproduced the entire "Ode to The Memory of Lady Harriet Elliot" below:
"Propitious heaven, her blooming virtues spare, Connubial love, condoling friendship cry'd; Fraternal fondness with desponding air in speechless anguish weeping at her side. Death, half-relenting at their pleading tears, approaches calm with dubious step and flow, a look like sympathizing sorrow wears, pauses awhile, then lifts his ebon bow. A diamond shaft dipp'd in those nectar'd springs, those streams of bliss that flow above the skies, he to his ebon bow applies, then mildly folds his sable wings, and with reluctance aims the momentary blow.
The vital-lamp more feebly burns, Ah, see its quivering flame retire; now it forsakes its station, now returns, and hovers there, unwilling to expire. The crimson-lustre of her damask cheeks, more vivid than the bright vermilion streaks with which the morning paints the eastern skies, now languishes, declining, pale, death o'er them draws his sickly veil, quick-throbbing at her heart, and swimming in her eyes.
The tender husband with extended arms while oft his lips her darling name invoke, fain would ward off from her devoted charms the fatal, decisive stroke; now wrings his hands, half frantic with despair, now hanging silent o'er the dying fair, soft from her clay-cold brow fate enthron'd e'en then in full triumphant state, wipes the presaging damps away: Oh! stay, my much-loved Harriet, stay! And must thou leave me here to mourn? Must thou so quickly take thy flight to thy own native realms of light, never, ah, never to return!
Fain would the voice of love-impassion'd woe detain her heaven-departing soul below. How shall it falter its last sad adieu? How disengage its fondly-lingering view from that dear form where it delights to stray, and where e'en life could gaze itself away? Yes, fled forever is that balmy breath! Cold, cold she lies! yet charming e'en in death! So looks the musk-rose, rooted from its bed, pallid, yet glittering with the morning dew; and so the new-blown lily droops her head beneath the fury of the northen blast, derang'd her foliage, dim her brilliant hue. Torn from her parent stem at last, the swain with grief beholds her lie, condemn'd to wither and to die; yet does he, pleas'd, her fainter sweets inhale, and own her still the beauty of the vale.
And art thou gone, ah nymph belov'd in vain? Too inauspicious, too malignant hour, when gloomy Atropos [one of the three Parae, or Destinies], relentless maid, disguis'd, in beauty and in joy array'd, mingled with Hymen's [the goddess who, according to the ancient mythology, presided over child-bearing] festive train; insidious revel'd in thy bridal bow'r; and while ascends the mix'd harmonious strain of social triumphs, happiness and love, with envious secrecy and utmost care twin'd with a branch of dark funereal yew and tarnish'd cypress shedding baleful dew, the smiling roses of the gay alcove: The poisonous drops its blushing charms impair, and quickly blast each infant blossom there But thou whose Muse can horror's powers command, Oh come, and picture the tremendous scene, when with Lucina, hand in hand, with stern inexorable mien, she issued from her sullen cell below, and hurrying to the beauteous victim's bed, rebuk'd unwilling Death's suspended blow, spread her remorseless shears, and clipp'd the vital thread!
Nor rank, nor worth, nor excellence could save the charms of HARRIET from the o'erwhelming grave, But thou, sweet babe, whose dear yet fatal birth, to death's cold arms thy hapless mother gave, May'st thou survive, with childhood's artless smile, alluring blandishments, and prattling mirth, A father's sorrows to beguile, to soothe the tender pang to rest which memory wakes to wound his breast.
Ye angel powers who innocence befriend, Let the lov'd pledge your choicest blessings share, from harm her guiltless infancy defend, and kindly make her [the baby daughter] your peculiar care! May she her [mother's] loss with due submission feel when ripening years shall teach her heart to mourn, when from the crowd she, sadly-pleased, shall steal to drop the duteous tear upon her parent's urn.
Why did we weep? Has the rude hand of Death Defac'd and blasted all that was so fair? No, she but seem'd to yield her breath; She lives, she reigns, she breathes immortal air! Attending angels caught her spotless soul, and bore it soft upon their silver wings to that bright seat above th' ethereal pole, the glorious palace of the King of kings; to wear a crown whose never-fading blaze far, far the starry firmament outshines upon essential excellence to gaze, that beauteous sun whose lustre ne'er declines, whose pure, unclouded, boundless-streaming ray through heaven diffuses everlasting day! Exulting through the crystal doors they flew, and as they mov'd towards th' ethereal throne, a cherub in a robe of azure hue, compos'd of woven undulating light (the sapphire's vivid beam not half so bright), Grac'd with a flowing, star-bespangled zone, eager advanc'd; upon her head a rainbow winds its orient wreath, a golden cloud her feet beneath. Her ruby lips ambrosial odours shed, as thus soft-opening, they benignly said: "My Harriet, hail! My sister and my friend: Come, share with me delights that never, never end! My heart was thine on earth, but here I glow with holier flames of mutual love; Thy mortal sister once below, Thy angel sister now above!"
She spoke; and speaking, round her Harriet's brows a fragrant garland elegantly twin'd, where amaranth and palm their bloom combin'd: then led her to the throne where heaven adoring bows! Where, plung'd in raptures at th' Almighty's feet, cherubs and seraphim in union sweet Triumphant hymn eternity away while all the emerald domes resound the choral lay! Dulwich-College. J.N. Puddicombe." 
1. Lord Sydney to his son, John T. Townshend. 29 September 1786. Nottingham University. Hildyard MSS THF/X/3/5, f. 11.