(2r2+1)2 where if r be put =1, the numbers come out 1 and 36 8r3 other numbers may be found by taking r=2, 3, &c. Solution to Question IT. = Put 4r+5+5 the product and 4r252 the difference of the required numbers where three conditions are solved, the difference a square, and the product plus and minus the difference squares, it remains to make the sum a square. But 16r+5+16r+5+45 is equal to the square of the sum; that is, 452 × (4r*52+4r2+1) must be made a rational square and its root a square, by proceeding as in the above Solution, we obtain 5 = (2r-1) and by taking r=1, the numbers come out 8r3 answers may be obtained by assuming r = 2, 3, &c..... Solution to Question III. and = other Put a x and x for the required numbers, whose product a2x2 a square, then a x2+a2x+x and a2x2-a^x-x, must be both rational squares, by taking the difference and reducing it to factors, we obtain the m root of the greater square —— + + m being any number what ax am m Za ever, by equating the square of this root to a2x2 + a2 x + x a2x2x(m2-1)/aam+m 1+m)2; ; make m2 n2, then 2a anx m 2a 2 and m, the numbers come out 125 and 125, which answer the condrtions of the question, various other answers can be deduced from the ge neral value of x. Solution to Question the 4th. Using the same notation as in the last Solution, it is required to make a2x2+ax-x and a2x2-a2x+x both squares, by resolving the difference into two factors, and equating the square of half the sum of said factors to a2x2+a2x-x, the value of x comes out (aa-1) m2 20 (aa-1)Xm2 20a and a x equal where mp21, and a any number at pleasure, unity excepted. If m = and a = 2, the numbers are 1⁄2 and 25, which answer all the conditions of the question. A PROBLEM. To find five squares the sum of every four of them a rational square, Dublin, May 18, 1812. Note The language of Algebra is very troublesome to Compositors, therefore the Proprietor of the Irish Magazine requests his Mathematical correspondents will simplify their operations as in the Companion and other English diaries. The following Verses were occasioned by a late whimsical Circumstance which occurred in the Smoaking-room adjoining the House of Commons. Some Wag chalked a Cross on the hat of the Right Hon. Pat. D Privy Counsellor was in a slumber. OLD jolly Sol's meridian beat Threw Earth into a muck of sweat, From the lax pores of every nation, He fucked the oozy exhalation. Andrew from dunghill, ditch and bög, The musky vapour, ftyl'd a fog, Which rifing on the breeze of even', As dark as hell had fmoak'd the heaven; To give my thoughts a different guife, The shoeboy Sun had ila ked the skies. Save where the pewter-rifaged moon In bright rotundity was fhown; Or where the star's nictating (park, Flared forth to prove the rest was dark. Thus have I feen (and I'm an odd fish) In gloomy pantry, fprats and cod-fish: To dim whofe rays, tho' fhades confpire, Greafy with ftreams of mucid fire. Or, what perhaps, better fits my knuckle, Seen Sheriff Harty's bright fhoe-buckle; Whose littering honours richly fet, Blazed brighter by being laid on jet. In fhort the dark to bring to light, And stifle metaphor 'twas night!!: Collected was that dread convention Which rules the roast of place and penfion; Which gives my Lord his princely houfe, and Snug finecure of "twenty thousand." And fills the laurelled Soldier's gullet With twelve-pence, and a-glorious bullet The Parliament in fine affembled, The dome with vocal thunders t:enabled; That vaftly dome, whofe Irish oak head Liftens alike to wit and blockhead; while the Where pother' echo gets fuch tearing, But if for honest famè yóû're eager, Which joined to Jack's narcotic notions Formed for poor Pat strong fleeping potions. 'Til his (quat length stretched on the bumboard, The Surregate of Ireland lumbered When lo, fothe gracelets deep-invader And Original Poetry. And as across the room he strutted, I fee you've got of words a throat-full, ON PATRICK'S DAY. On Patrick's Day, no matter where, Then hireling minions ne'er shall wave In focial draughts fhall downward glide, Irish dan What pity then, fuch louis as thofe SONNET. AS at eve, o'er the woodlands I foray, The fweet feather'd chorifters fing. How vocal, harmonious and fweet, Yes, Anna, the pride of the plain, The fairest of all that are fair, Alas! I am doom'd to despair. I feat that a more happy fwain, 'Tis madness to think we must part, FOR THE IRISH MAGAZINE.. The underneath Verses on the subject of the intemperate proceedings at the late meeting of the truly "contemptible and obsequious Corporation of Dublin" having appeared with yarious alterations in an Evening Paper of of this City, we here (by request) publish them in their original form. -000 AIR-"GARRYONE." 1. OH! the heart that's by ruthless intolerance led, For dread soe'er our fortunes be, If we're rolled by mercy's sacred sway, Oh then Irishmen heed not the malice of knaves, 2. Whilst we blame the poor dupes, the poor fanatic crew, Oh! his morals 'tis, and not the man, That will share th' abhorrence of the just, Bear love to her sons, tho' unworthy, we must. O then Irishmen heed not the malice of knaves, Nor For the wages corruption bestows on her slaves, Cannot equal the pain which remorse leaves behind! * Vide Lord Donoughmore's Speech on the Catholic Question. A NEW A NEW SONG, composed by the Rev. Guideon Ousley, Missionary Extraordinary and legal successor to the Rev. J. Wesley.-Air, Conny O'Dignan. I sing you the type of ill-will and disunion, Sir, His teeth are worn out in the bone of contention, CHORUS. So cut away, dash away P-y O'D—g-n, He howled for Fitzgibbon, as cur dogs in ire would do, Chorus, So cut away, &s. With tortures and racks and bastiles he annoyed us, Rye-house and meal-tub and gun-powder plots he knows, Houghers, united men, croppies, and-SANS CULOTTES-- |