294 



Mr. E. W. K. Edwards. 



[Jan. 28, 



Principle of the Area-Scale. 

 Suppose the equation of a curve to be 



r 2 = a + bB + cW + dP, 

 the area between the curve, the initial line, and the radius vector 



C29 



making angle 2(9 with the initial line, is J r 2 old • i.e., 



"Jo 



a + 2bd* + f c<9 3 + 4d<9 4 , i e., S, say. 



Now if ?'o> be the outside radii of the sector so quadrated, and r\ be 

 the radius vector bisecting the angle between them, it can be seen that 



?- 2 = a, 



n 2 = a + bd + c&. + dP, 

 r 2 2 = a + 2bO + <Lc& + 8d6S; 



whence, assuming (Ar 2 + B?i 2 + Cr 2 2 ) = S, and solving three of the 

 four simultaneous equations for A, B, C, we get values A = B = §, = J , 

 which also satisfy the fourth. 

 We have then 



S = }6 (r 2 + 4r 1 2 + r 2 2 ). 



The design on this particular transparency is made on the assump- 

 tion that each of the ten separate portions of the curve between the 

 first and third, the third and fifth, . . . the ninth and eleventh 

 radiating lines, approximates to some member of the above family of 

 curves. 



Anyone acquainted with the discussion of areas, in Cartesian co- 

 ordinates, in terms of a series of equidistant ordinates, and their 

 common distance apart, given in Bertrand's Calcul Integral, Section 363, 

 will see an analogy in the foregoing expression of the area of a 

 sector in terms of the squares of three equally inclined radii and 

 their common inclination. Other designs of the transparency can be 

 made on the assumption of larger portions of the curve approximating 

 to members of families of curves in which r 2 is equal to a rational 

 algebraical function of in ascending powers of 6, of degree higher 

 than the third. On the whole it is easier, and possibly more effective, 

 to use the method employed above, in the same way as " Simpson's 

 Rule " uses the corresponding theorem in Cartesian co-ordinates. 

 There is no reason to suppose that " Simpson's Rule " gives a less 

 accurate approximation in the generality of cases than "Weddle's 

 Rule " * or the numerous rules deducible in terms of the co-efficients 

 calculated by Cotes. 



An investigation of the family of curves 



r 2 = a + bd + c&t + d&t 

 * See Boole's ' Finite Differences.' 



