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The phrase the long arm of the law, and its variants, designate the far-reaching, inescapable or punitive power and influence of the law—especially, in later use, as represented by the police.

—Note: More generally, the noun long arm has long been used, in the singular and in the plural, to designate far-reaching power and influence. The following, for example, is from Vertues Common-wealth: Or The High-way to Honour (London: Printed for Iohn Newbery, 1603), by Henry Crosse [sig. M]:

Many wealthy Yeomen, & rich Farmers that are risen vp to goods inough […] are eyther couetous of some vaine-glorious title of gentilitie, or otherwise so miserly greedie of wealth, (for one of the two I know not which) thrust themselues […] into Cities, Corporations, and Liberties, and yet holde theyr Farmes still in their owne occupying: for they haue such long armes, that they claspe many great liuings.

These are, in chronological order, the earliest occurrences of the phrase the long arm of the law and variants that I have found:

1 & 2-: From two texts by the English merchant, author and philanthropist Jonas Hanway (1712-1786):

1-: From a letter to the Editor, published in The Public Advertiser (London, England) of Monday 14^th December 1767 [Letter 13, page 2, column 2]:

Liberality and Charity are too very different Things, but equally necessary. Charity puts an Axe to the Root of Extortion; but Trade, which promotes Liberality at the same Time, shows us how Commerce, in its most advanced Stage, militates against the Virtue and Religion of a People; and how difficult, if not impossible, it is even for the long Arms of the Law to prevent mutual Oppression. Avarice, Ambition and Excess, bring on the very Evils we most wish to shun—and Vice and Slavery tread close at their Heels. Liberty is wounded in its vital Parts whenever we commit any Act tending to oppress the Indigent, or violate the common Rights which Heaven has ordained for the peaceful Enjoyments of Mankind in their several Stations.

2-: From Virtue in humble Life: containing Reflections on relative duties, particularly those of masters and servants. Thoughts on the passions, prejudices, and tempers of mankind, drawn from real characters. Fables applicable to the subjects. Various Anecdotes of the living and the dead: In Two Hundred and Nine Conversations, between a father and his daughter, amidst rural scenes. Intended as an amusing and instructive library to persons of certain conditions, and proper for all families seeking domestic peace and Christian piety: with a Manual of Devotion, comprehending extracts from the scriptures, prayers, hymns, religious poems, &c. (London: Printed for Dodsley, 1777) [Volume 1, Part 2, Conversation 3, page 94, column 1]:

Your mistress is so exact, she makes up her accounts every Monday morning, paying ready-money for every thing the buys. By this means, she lives elegantly and splendidly, for half, or at most two thirds the sum, which those spend, who would never pay their debts, if they could cut off the long arms of the law. Her maxim is, “better go to bed supperless, than rise in debt:” intimating, that many charges are contracted by things not necessary; and that some necessary things must be given up, rather than run in debt for them.

3-: From The London Chronicle (London, England) of Tuesday 16^th February 1802 [page 158, column 2]—the noun crim. con. is an abbreviation of criminal conversation, designating unlawful sexual intercourse with a married person:

On every ground of morality, of policy and justice, we trust the report of the permanent establishment of a Private Theatre is unfounded; or, if not, that the long arm of the Law will be able to reach it. There have not been wanting instances of late years of the ruinous effects of Private Theatricals. It is known to numbers that they are the very hot-beds of seduction and crim. con. If such entertainments are to be supported by Subscription, they certainly come within the letter of the Law, for it is the same thing whether persons pay for their admission by the night or the season.

4-: From The Monuments and Genii of St. Paul’s Cathedral, and of Westminster Abbey; with Historical Sketches and Descriptions of both Churches: Forming an entirely new and correct Biography of all that is interesting in the Lives and Achievements of the most illustrious Characters of the United Kingdoms (London: Published by John Williams, 1826), by George Lewis Smyth (1800-1853) [Volume 2, Oliver Goldsmith, page 524]—Oliver Goldsmith (1728-1774) was an Irish author; there is a memorial to him in Westminster Abbey’s Poets’ Corner:

He had never the heart to hear the prayer of distress and refuse to relieve it; and when called on for assistance, as he often was by his poor countrymen, if he had not money, he would give away his clothes, and desire them to turn the only things he could part with into the direct substance of their desires. This beneficence of disposition early involved him in difficulties. A fellow-student prevailing on him to become security for the payment of a tailor’s bill, he was soon obliged, in consequence of his inability to keep the engagement, to leave Edinburgh precipitately. But the tailor pursued him in his retreat with the long arm of the law; he was arrested in Sunderland, and conducted back to the college by bailiffs.

5-: From a transcript of the speech that the British politician Ralph Bernal (1783-1854) delivered in the House of Commons on Friday 24^th May 1833, published in The Saint James’s Chronicle, and General Evening Post (London, England) of Saturday 25^th May 1833 [page 2, column 2]:

At various times resolutions had been come to by the House of Commons for preventing and punishing bribery. […] These laws were good, but he was sorry to say that they were not carried into effect. The House treated bribery cases too negligently. They coquetted with them, but never approached them in a manly and steady way. It was a pity that the strong and long arm of the law was not exerted.