TOL. XLVIII.] PHILOSOPHICAL TRANSACTIONS. 471 



X, by f), q, r, s, &c. z"" — 2i/z" + I will be = {z" — 2pz + \) X (z' — 2^2 

 -\- 1) X {z' — 2rz + l)> &c. («), when n is a positive integer, (as we shall al- 

 ways suppose it to be), let z be what it will. 



Hence may be easily deduced a demonstration of that remarkable property of 

 the circle first discovered by Mr. Cotes : but as that property has already been 

 demonstrated by several mathematicians, Mr. L. omits taking any further notice 

 of it, and proceeds in the investigation of soiiie other useful theorems which had 

 never been published. 



^rt. 2. — If j/ be = 1 ; then, a being = 0; p, q, r, &c. will be the cosines of 

 ?-, -, --, — , &c. (n) respectively : therefore /> will be = 1 ; and, if n be an even 

 number, one of the cosines q, r, s, &c. will be := — 1 , one of the arcs 

 £ £L — &c. being then = ^. 



y/r/. 3. — If?/ be = — 1 ; then, a being = r 5 P) ?' ^^ *> &^- will be the co- 

 sines of—, — , -T-, &c. (n) respectively: therefore, if n be an odd number, one 



c 

 of those arcs will be ^ , whose cosine is — 1 . 



yJrt. 4. — If, in the equations z'" — 2yz" +1=0, and z^ — Ixz -\- 1=0, 

 we substitute « — 1 for z, they become {v — xY"" — 1y X {v — l)" -\- 1 =0, 

 and {v — \y — 1x X (f — l) + 1 = v" — (2 + 2jr) X v + 1 + 1x = Q. 

 Consequently 



On 1 ~ 



i;'" — inv"-"-' + + -In X — — - v" — Inv +1 



. . . . -f- 2yn X — -— v^ + 2ynv -f- 1y 



(y'' — 2 + ip X V + 1 + 1 p) X {v" — 1 -\- 1q X V + 1-\- 1q) X {v" — 2 +2r 

 X y -|- 2 + 2 + 2r) X &c. (n) ; where, of the two signs prefixed to the terms 

 where ^ is a factor, the upper or lower takes place, according as n is an even or 

 an odd number. Whence, by the nature of equations, it follows, that (2 + 2p) 

 (2 + 2q) X (2 + 2r), &c. is = 2 + 2y. But this equation vanishing when y 

 is = 1 and n an even number, or when 3/ is = — 1 and n an odd number, it will 

 be proper to consider those two cases more particularly. 



Jrt. 5. — First, let us suppose 7/ = ], and n an even number: then p being 

 = 1, and one of the other cosines q, r, s, &c. = — 1 (Jrt. 2) ; we shall have , 



D^" — 2wt;'"-' + + w' i;^ = (v' + O) X (v'' — 4i; + 4) X {v'' — 2r+2q) 



X {v + 2+ 2q) X {v'—2 + 2r X V + 2 + 2r), &c. Therefore dividing by v'', 



v"^-" — 2m/'^M- +n' ={v^ — 4v +4) X {v^— 2 + 2q X v + 2 + 2g 



X (u'^ — 2 + 2r X V -\- 2 -\- 2r), &c. that factor in which the value of the co- 

 sine q, or r, &c. is — 1, being expunged. 



