446 Mr. S. M. Drach's easy Rule for 



'^=p<^ + qfr{c±s)i-a{c±s)\-\-h{c':\isYp 

 rP+?{(c+s)r' + (c-s)r*} =2rP+*cos (i? + y)^ =2aP+«cos q^^ 



1 . 2 . . 2 



[_ + (c — 5)p^+9^-''' (c + s)!* = 2 cos r^'iZ — ^V J * 

 As before, de^^elope 



2 cos ( ^ + g) 9 = 22' A . (cos 9 = ^ j \ 



and put it equal to the sum of 



2 cos (^-i)rl;' = 2B,(2 cos 4;'= '^'f'^' = ^)\ 



each having its binomial multiplier for the general equation 

 of the curve, in both cases. 



Ex, w=-3, p + g'=4, q=l', .'. 16^'^-16^V2 + 2^-4=^ 



+ 8ba^ + 6baQ+ ^ (Q2-2a2Z>2^+ ^ (Q^-Sa^i^Q). 



All these examples verify a previous individual and trouble- 

 some method of eliminating cos <p from the original equa- 

 tions. 



Generally 



^=rcos9, r^— «^— 5^=2a5cos\I/; 



and for p positive, 



rP-^. 2 cos {p-q)Q = {bK° + aK)P-'iy K'=2 cos (p-i)^; 

 p negative, 



rP+1. 2 cos {p + q)Q={aK^ + bK)P+9^ K»=2 cos(5'-2>, 



an easily memory-retained formula, to be developed by the 

 binomial theorem, akin to the finite difference series 



i/.= (l+A2/o)^. 



The final equation with p negative, p + </ = ^, being 



r^.2cos^& = {aK^-i-bKf^ 



the left-hand member is 



F{F--j-l){F-J-2)..{V-2j+l) 

 - 1,2.3, .J ^ ' 



