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Demonftratlon — Let a, a', «", d\ &c. reprefent any arcs 

 J, s, j", / , their fines 

 c, c\ c, c", their cofines 



Then by the common theorem for the fiine and cofine of the 

 fum of two arcs. 



The fine 



The cofine 



The fine t „ . , „ ^sc c -]- s cc -\- s" c c — s s s 



The cofine 



&c. &c. 



•) ^ , CSCJU s c 



{oi a + a =1 ] , ^ , 

 J (-cc — ss 



■J ^ ^sc c 



( of a + a' + a" = } , „ 

 J cc c 



The following obfervations may be readily made by confidering 

 the way which in thefe fucceffive values are formed. 



1. In both fine and cofine of the fum of n arcs [a-^ a -V a" &c. 



the number of factors s s cc in any term is equal to n and 



that the fines j, j', /, &c. and alfo the cofines c, r', /, &c. are 

 concerned exadlly alike in the whole quantity. 



2. In the fine of the fum of n arcs (« + «'+ &c.) the greatcft 

 number of cofiues c, <r', /, &:c. together in any term = 7i — i . This 

 numl- er diminiflies by 2, and confequently the number of /, /, &c. 

 increafes by %. 



3- In 



