[ 3^ ] 



3- In the cofine of the fum of n arcs the greateft number 

 of r, c\ c", 5cc. in any term — n the next lefs number n — 2, &c. 

 and confequently the number of /, j-, &c. increafes by 2. 



4. With refpe6t to the fij»ns of the different produfls — In 

 the fine of « arcs a-\- a -\- a -^r ?fc ) when i, 5 or4^ + i (/ being 

 any number /, /, /' &c. are united together, the fign is + other- 

 wife — . In the cofine of « arcs when 2, 6, 10 or 2/ (/ being odd) 

 J, /, /' are united together the fign will be — othcrwife +. 



5. In no term can the fine and cofine of the fame arc 

 occur. 



6. In any term s r r - - - c c c - - - whether of the fine or 

 cofine if m be the number of the cofines and confequently tn — n 

 the number of the fines : then, becaufe each of the quantities 

 J, s' &c. and alfo r, c &c are concerned exactly alike in the 

 fine of the fum of n arcs [a -^ a -\- a + &c.), and alfo in the 

 cofine of the fum of n arcs {a-^ d -^ a + &c.) and likewife be- 

 caufe the fine and cofine of the fame arc cannot occur in the fame 

 term, it follows that the number of terms x//' (w terms) - - - 

 c c c' \jn — n terms) = the number of combinations of « things 



taken m together =: «. n — x. n — 2 - - - « — m — r 



I. 2. 3. — - m 



From 



