408 Mr DAVIES on the Equations of Loci 



r d & T d 



or writing for sin tan" 1 j - and cos tan" 1 -j - their values we get 



a coa.6 { r,d 6, cos 6, dr, sin 6} + sin 6 {dr, cos 6, r,d 6, sin 6,} ,gg . 

 ~~ ~~ 



_ r, d 6, cos 6 6, d r, sin 0, (g3 . 



VT^Ttf/' 

 Also from (20) we have 



~ 





cos 6 6, sin 6 6, dr, 



r. r. r.dd 



6. To find the equation of the normal. 

 Here the angle HOF is denoted by tan" 1 - , or 



= - tan' 1 



Resuming the general form of (19) expanded, viz. 



= cos /3 cos 6 + sin /3 sin 0, 



we have, by substitution, 



which inserted in (23) gives 



l_ r,d6,cos6 6, dr,sin6 6, / 25 ^ 



and writing for sin tan" 1 -~^ and cos tan" 1 ^- these values, we ob- 



tain 

 _0___ cos 6 { cos 0, d r, sin 6 l r,d6 l } + sin { cos 6, r,d6,+ sin 0,drJ 



... (28.) 



