of Cotes* s Theorem. 13 



Lastly, if the angle MSP = — , we shall find, provided n be 

 of the form 6w + 1 , 



SP....SP = L" {1,7}. 



I n 



Let US now resume the general equation {iji} and first, 

 let 9 = 0, or, let one of the / terminate in the second vertex, 

 we have, for every value of «, ' 



J^^ ^(») — a" (^+^*)" __ ^« (^+^-"')" (2py (COS. «)" 



{1,8}. 



2. Let cos. w5 == — 1, and we obtain 



JO J0__^« 0+!^__^« fi-FX— ')" __ i^py fcos. «>» 



(COS. — 



— ' ^ 



{1,9}. 



This embraces all the cases where n is of the form (sw + 1) 



-J, and among the rest, when n is any odd number, and one 

 i 

 of the r terminates in the first vertex ; when n is of the form 



4,w + 2, and one of the r perpendicular to the axis, &c. 



3. Let cos. n9 = 0; then 

 I 



JO J*^ — /,» (i+x')" _ n (>+^-'>' _ (2/>)" (COS. »y c 1 



This includes the cases where n is of the form (sw + 1) . 



(f) 



~-^, as for instance, where one of the r^*^ is perpendicular to 



