470 PHILOSOPHICAL TRANSACTIONS. [aNNO 1754. 



fluents more easy, will, on perusal, be obvious, Mr. L. says, to every one ac- 

 quainted with that branch of science. 



Art. 1 . — Supposine '^=- = — » where x and y denote the fluxions of 



the variable quantities x and y respectively, and n an invariable quantity ; it is 

 proposed to find, in terms of^ and z, the equation of which z is a root, and z^ 

 — 2.rz + 1 = O, a divisor. 



Taking the fluents of the given flux iona ry equation, we have, supposing x = 1 

 when y is = 1, hyp. log. of {x + y^x^ — l)" = hyp. log. of y + \/y^ — i, or 



(ar + Vx^ — \y =y + '^y'^ — 1 : whence, by substituting for x its value— -^^ 



(found by the equation z' — 2xz +1=0), we have z" = y -\- ^ y~Iir\ •_ there- 

 fore z" — yis = ^y"^ — 1 ; and, squaring both sides, z^" — lyz' + ^^ = ^' _ i. 

 Consequently z^' — lyz' + 1 is = O; which, supposing n a positive integer, is 

 the equation sought. 



Now it is obvious, n being such an integer, that this equation will have as 

 many trinomial divisors, of the form z' — Ixz -|- 1, as there are values of x corres- 

 ponding to a given value oiy: which values of .r, when y is not greater than 1, 

 nor less than — 1 (the only case I propo se to consider), will not be readily ob- 

 tained from the equation {x -\- »/ x^ — Xf ■=y -\- V y- — 1 found above: but, 

 if we multiply the given fluxionary equation by , we get — -— — = — ^ — ; 



of which the equation of the fluents is n X circ. arc rad. 1, cosine x = circ. arc 

 rad. 1 . cosine y ; where a; is := 1 when y is = 1 , agreeable to the supposition 

 made above, when we took the fluents of the given fluxionary equation by loga- 

 rithms. Therefore if a be put for the least arc whose cosine is y, and c for the 

 whole circumference, radius being 1 ; y being the cosine of a, a -j- c, a -j- 2c, 



A + 3c. &c. X will be the cosine of ^, ^t_E, L+_2£ &c to Lti^JZ^JiS, 



Consequently, expressing the last-mentioned cosines, or the several values of 



motion, an investigation of the naotion of the equinoxes, in which Mr. L. has, first of any one, 

 pointed out the cause of Sir Isaac Newton's mistalie in his solution of this celeb; ated problem. Mr. 

 L. as he had chiefly completed this work during some intervals from the stone, with which disorder 

 he was severely afflicted in the latter part of his life, so he just lived to see it printed, and received a 

 copy of it the day before his death, which happened Jan. 15, 17i)0, at Milton, near Peterborough, 

 being 7 1 years of age. 



About the year 1762, Mr. L. became agent and land-steward to Earl Fitzwilliam; an employment 

 which he resigned only 2 years before his death. And in 1766 he was elected f. u. s. Though Mr. L. 

 was doubtless one of the greatest mathematicians that this or any country has produced ; his merit in 

 this respect was not more conspicuous than his moral virtues. As his compositions were profound, and 

 elegantly clear and simple ; so his manners and deportment were manly, dignified, genteel, and be- 

 nevolent. The strict integrity of his conduct, his great humanity, and readiness to serve every one to 

 the utmost of his power, procured him respect and esteem from all his acquaintance. 



