In the first-instance judgment  2 Q.B. 484, Hawkins J wrote "The facts not being in dispute." But how could've Carbolic Smoke Ball Co trusted Ms Carlill's proper use? Perhaps she misused it, or lied about using it at all after buying it!
Mindy Chen-Wishart. Contract Law (2018 6 edn). p. 60.
An advertisement may also, exceptionally, be treated as an offer rather than an invitation to treat. In Carlill v Carbolic Smoke Ball Co (1893), the manufacturer advertised the ‘carbolic smoke ball’ and offered to pay £100 to anyone catching influenza after using it in the specified manner, adding that £1,000 had been deposited in the bank to show their ‘sincerity in the matter’. C successfully sued for £100 when she caught influenza after proper use of the smoke ball. The court rejected the manufacturer’s claim that the advertisement was too vague and not seriously made. The Court held that: (i) the advertisement was an offer to the whole world; and (ii) a unilateral contract (see 2.3.6) was made with those who met the condition ‘on the faith of the advertisement’.
Ewan McKendrick. Contract Law: Text, Cases, and Materials (2018 8 ed). p 55.
The defendants advertised their medicinal product, the now infamous ‘carbolic smoke ball’, in various newspapers in the following terms:
‘100l. reward will be paid by the Carbolic Smoke Ball Company to any person who contracts the increasing epidemic influenza, colds, or any disease caused by taking cold, after having used the ball three times daily for two weeks according to the printed directions supplied with each ball. 1000l. is deposited with the Alliance Bank, Regent Street, shewing our sincerity in the matter. During the last epidemic of influenza many thousand carbolic smoke balls were sold as preventives against this disease, and in no ascertained case was the disease contracted by those using the carbolic smoke ball. One carbolic smoke ball will last a family several months, making it the cheapest remedy in the world at the price, 10s., post free. The ball can be refilled at a cost of 5s. Address, Carbolic Smoke Ball Company, 27, Princes Street, Hanover Square, London.’
The plaintiff, in reliance upon this advertisement, purchased and used the product as directed but subsequently caught influenza. She sued for payment of the £100 and succeeded before Hawkins J. The defendants appealed to the Court of Appeal but the appeal was dismissed. It was held that the terms of the advertisement constituted an offer, the terms of which were accepted by the plaintiff with the result that she was entitled to recover the promised £100.
Bowen LJ [I skip the judgment extract.]