[ 34 ] 



Demonstr. By fubftituting in tlie values of the fine and 

 cofine of n a found by the laft theorem, for n fucccffively 2, 3, 

 4, &c. and exterminating s it may be conje»ftured that the ge- 

 neral terms of the fine and cofine will be as here ftated. That 

 this conjedure is true appears in the following manner : 



-I — u 



Let Be- be a term in the cofine of n — i ^, and Cr 



n — u — 2 



i/i — f-, and Dc v/i — c'' terms in the fine of « — i a ; and that 



the latter terms will be of this form appears from the former 

 theorem. Applying the common theorem for the fine and cofine of 



n — u 



the fum of two arcs, it readily appears that the coeff. of c in the 

 cofine of « ^ = B — C + D. 



Now fuppofing the theorem generally true and fubfiituting 

 in the general terms for w, n — i and for u fubft. u, u — i and 

 u + I fucceflTively, the refult is 



u 



n—u—z n — r- n — «. n — « +1 to — terms 



B= + 2 X 



2 



„_„ n—ii + I. n—u -\- 2 



C=±a X 



u 



to — terms 



u 



to 1 terms 



2 



,^^ to J terms 



D = 



