of Cotes* s Theorem. ig 



= ^ • -f : ^^ fi.or 



j I— 2?,^" (2 . COS. «?)*— l) -i-M" Cr 



R R = ^'* • 1 TT . . . . j 7,3 1 • 



I X ] I— 2^*'• . COS. 2np+x*" C* t ' ^^ i 



. Let <t> = 0, or, let the extremity of one of the R lie in the 

 I 



principal vertex. If n be even 



Rr) n (I— x'')" « (sin. «y S^ m\ 



1 „ (^^^ (sm. 7«)* ^ -> 



If odd, 



Rr> n (1— X*)" n (sin. *y I ^ /- 1 



i—X*" sin. no, L J 



I n 



Let <p = ~, and ;z = 4W + 2, In this case, two out of the 

 I 



R are at right angles to the axis, and 



R ^=^-m i7.6}. 



in 



Again, let one only of the R be perpendicular to the axis, and 

 R R = '^"-^-|^ ; • • • {7.7}. 



in * 



Here n is of course odd. 



Next, let one of the R be inclined at an angle — , to the axis. 



If w = 4w + 2, 



R R = '^"-Hi^- • {7.8}. 



in ^ •' 



and it is curious to observe, that this expression is the same 

 function of a, x, n, as that of 1 7f7l» 

 Un be of the form swi + 1, 



« ^=^"-m M- 



Lastly, let n be of the form 6m + 2, and <p = ^, then 



Da 



