[ 5 7 s ] 
(found by Art. VII.) a — a for &>, the fame becomes 
- — n n ~ n — . ■ - — 
2 a — a — 2 y u x 2 a — a + u — 2 i y y. 
a 2 — 2 a a +- a 2 -j- b % x a 2 — - 2 an + a% + c% i &c. 
= 2 + 2 y x a 2 — 2 a u + '/&* x a 2 — 2 a a + > 2 * 
a* — 2 a u + J 2 , &c. if inftead of V a 1 + b 3 , 
Va 2 + &c. (the fecants of the arcs of which 
b, c , d , &c. are tangents), we put /3, jT, &c. 
And, by a like fubftitution in the equations in 
Art. XI. and XIV. it appears, that 
-• » . 
ia — u — u is = 2 a ax a — u x a 2 — 2 aa-f-> 2 
xu* — 2 a a +• 2 , &c. or 2 x a — ax a 2 — 2 au-\-y z 
xa 2 — 2 an +• cf *, &c. according as a is an even or 
n n 
an odd number : And that 2 a — u + « is = 2 a a 
x a 2 — 2 a u + ( 6 Z x a 2 — 2 a a + > 2 > &c. or 
2 xa 2 — 2 a a + /3 a x a 2 — 2au y z y &c. accord- 
ing as n is an odd or an even number. 
From what is done above, I might now deduce 
many corollaries j and, by means of other fubftitu- 
tions, inveftigate other theorems 5 but want of leifure 
obliges me to defilt. 
LXXVII. An 
