[ 49 ] 



I. ift. When n isof the form \p + i, fubft. for », 4/ 4- i 



cs n — 2 m. Q — a— cj, 4/ — 2 »? O^ 4- Q — n — zma = (becaufe 

 adding or fubtracfting the circumference makes no alteration in 

 the value of the cofine and adding or fubtradling \ the circum- 



ference changes the fign of the cofine) + cs O; — ?i — 2 m a ~ 

 +_s, n — 2 ma -\- when m is even and — when odd. 



I. 2. When n is of the form 4/"+ 3, fubft. for », 4/> + 3, 



CJ-, n — 2 m. O; — ^ = cj, 4/> 4- 3 — 2 th Q, — n — 2 ot ^i = + /, n — 2 /« ^, 

 4- when m is odd-and — when even. 



2. I. When n is even of the form %p, /being odd, fubft. for« 



2/>, cs n — 2 m. O; — a — cs ip — 2 m Oj— « — 2 « a ^ + cs n — 2 w <r, 

 4- when ffz is odd and — when even. 



2. 2. When « is of the form, \p; fubflituting for «, 4/, 



rX « 2 W. Q; — 1 — cs \ p 2 7H, O;; — « 2 W, <7 = + CJ « 2 »», <? 



4- when m is even and — when odd. 



Whence fubflituting in the general term for the cs, Q;— ^, the 

 J-, a and for cs, n — 2 m a, the values above found, the truth o£ 

 the theorem is evident. 



Vol. VII. G Theorem 



