560 REPORT — 1903. 



3. The Use of Tangential Coordinates} By R. W. H. T. Hudson. 



There are two reasons why it is advisable that a greater use should be made 

 of tangential co-ordinates in elementary analytical geometry. From an educa- 

 tional point of view they are useful in drawing out the student's power of deduc- 

 tion, and exciting his interest in a way in which the long and difficult problems, 

 with which our text-books are crowded, fail to do ; and, secondly, there are many 

 theories which find their most natural expression in these coordinates, chiefly 

 because the absolute has a less specialised form in tangential than in point co- 

 ordinates. For example, it is an easy exercise to express the equation of a circle 

 in the form 



kiP + m^) + {Gl + Fm -I- C)2 = ; 



and then, from this, the whole projective theory follows clearly. Again, to take 

 examples from more advanced parts of the subject, the /oci of the curve 



(^{l, m, n) =0 

 are given by the roots of the equation 



where z = .v + iy. The centre o( a curve of class v is best defined as the polar 

 point of the line at infinity, and has for equation 



8"-i^/a«-i=0. 



From this the property of being the centroid of the points of contact of parallel 

 tangents follows without further analysis. 



Great clearness is introduced into the theory of averages in connection with 

 areas and volumes by the exclusive use of tangential coordinates. The equation 

 of the miU-conic of an area is 



r/{lx -t- my + iif civ dy - 0, 



which may, by proper choice of axes and use of the notation of averages, be 

 written in the form 



Z' x'' -f wry- -t- n- = 0. 



Then the ellipse of yyration is the confocal 



I- y- + m-.v — n- = 0, 

 and the ellipse of inertia is the conic conjugate to the null-conic 



r.t- + m~y —n- = U. 



Finally, the surface of floatation is a good instance. In this case, as in other 

 cases of approximation, it is well to take the standard equation of a plane to be 



z + n = Lv + my 



so that s and n are small quantities of the second order ; and then the plane equa- 

 tion of the surface takes the elegant form — 



2n + ^ff{l'^ + ^^^f? fix dy = 0, 



dN 



the integral extending over the section of floatation and V being the volume 

 immersed. 



' Math. Gazette, No. 42, Dec. 1003. 



