396 Mr DAVIES on the Equations of Loci 



Reducing (2), we have 



sinp sin 4* IF cos p cos (f> = ^Fcos \; 

 from which, by differentiating, we get 



(3.) 



d^ _ _ sin (p cos p ^ . 



d (f) ~ cos 4 sin p 



__., sin (ft d (ft 

 Whence tan p = inr 



cos 4/ d 4- 



sln = 



cos p = 



-f- sin (ft d (ft 



cos 4 1 d A 



(5) 



. 

 V cos" 4- d 4,* + sin 2 



Insert these values of sin (ft, cos (ft in (3), and reduce ; then, 

 _, sin 4- sin <ft d (p + cos (ft cos >!> d ^ 



sinX=: 



cot X = 



(p cos 2 4* (d 4^ + d 2 ) 2 sin (p cos sin ^ cos 



cos 



? + sin 3 <f> d (p* 

 sin -^sm(pdd) + cos d> cos 4" d -^ 



V sin 2 ) cos 2 



2 sin <p 



..-,... (6.) 



whence from (5) we know the radius of curvature, and from (6) the polar 

 distance of the centre. 



By restoring the value of 4/ in terms of < Q, we should, of course, obtain 

 the several values in terms of the co-ordinates of the point of contact. This 

 purpose might, however, be easily effected by direct investigation, and we 

 should avail ourselves of that circumstance to form them into mutual checks 

 of the accuracy of our operations and results. Too much attention cannot 

 be given to these verifications, where, as in the present case, the formulae 

 are destined to become fundamental in respect to a long series of important 

 subsequent inquiries. 



Let the circle of curvature be denoted by 



cos p = cos X cos + sin \ sin cos 6 K (7.) 



and the curve by 



F((j>'&) = (8.) 



