190 Mr. Davies on the Employment of Polar Coordinates 



4. When lime and magnesia exist in combination with car- 

 bonic acid, the whole weight of the acid in combination being 

 previously known, to ascertain the weight of each base. 



Rule. — Multiply the whole weight of material experimented 

 on by 77 ; from the product subtract 147 times the weight of 

 acid, divide the remainder by 22, and the quotient will be the 

 weight of lime : — 



77 £ — 147 a -v. r.,1 r 



— = weight or the lime. 



22 & 



Crickhowel, July % 1842. 



XXXII. On theEmployment of Polar Coordinates in expressing 

 the Equation of the Straight Line, and its application to the 

 proof of a property of the Parabola. By T. S. Davies, 

 Esq., F.R.S., F.S.A., fyc, Royal Military Academy, Wool- 

 wich *. 



A BOUT ten years ago I gave in a note to my paper on 

 **■ Spherical Coordinates (in the Trans. Roy. Soc. Edinb., 

 vol. xii.) the general equation of a straight line in reference 

 to polar coordinates. The idea, which is very simple, was 

 suggested by the method which I had employed in the dis- 

 cussion of spherical loci ; the equation of the line in piano 

 corresponding to that of the great circle on the surface of 

 the sphere: and it was made apparent that the treatment 

 of the straight line by such means was quite as simple and 

 elementary in all its details as that by means of rectilinear co- 

 ordinates. 



Beyond the occasional employment of the expression 



d r 

 • ,.'- to express the angle of the tangent and radius vector, or 



the relation between the perpendicular on the tangent and 

 the corresponding radius vector, the method of polar coor- 

 dinates has been generally disregarded by mathematicians 

 in treating of the tangents and normals to curve lines : and 

 I do not recollect a single instance where the general polar 

 form of the equation of a line subject to its adequate number 

 of defining conditions has even been noticed, much less used, 

 by any author, prior to the appearance of my paper. However, 

 that it is a very efficient method of investigating the properties 

 of rectilineal figures, any reader may readily convince him- 

 self by a few experiments upon such theorems as express those 

 properties ; and I wish here to illustrate its utility in reference 

 to tangencies by the investigation of a theorem which has ex- 



* Communicated by the Author. 



