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laft term. The general term is 



tt n — I. « — 2 - - to ff? terms 



I. 2. 3 '- '- m 



cs n — 2 m a. 



Dem. Let a, a'y a\ &c. reprefent any arcs 

 c, c, c", &c. their cofines 



Then by trig, cs, a X 2 cs, a' ■= cs, a + a + cs, a — a 



andmlikemam'iercj-,flX2fj-,<z'x2CJ', fl" = rj-, a-]-a'-\-cs,a — a\ ics,a" 



= ex, a + a' + ci' + <rj-, « + d — d'-\-cs, a — a + al' + cs, a — a — a, &c. &c.. 

 and it is evident that to multiply by twice the cofine of any 

 arc it is only neceffary to encreafe and diminifli each of the 

 former quantities a -\- a -\- Sec. a — d, &c. by that arc, and take the 

 fum of the cofines of the arcs fo encreafed and diminifhed : therefore 

 becaufe in the produdl of the cofines of a, 2 d, 2 d\ &c. all the 

 arcs d, d', &c. muft be involved exadlly alike, it follows that 



2 ^ cs, aX cs, a' X cs, a' X &c. = fum of the cofines of all the 



arcs formed by adding to a each combination of the « — i arches 

 d, a , &c. taken pofitively or negatively. Hence by the theorems 

 for combinations, there will be 



I. term the cofine of a -f «' -f d' + &c. 



