Changes of Curvature by Means of a Flexible Lath. 413 



na + btx + ctx 2 + d%x* = %y, 

 a%x +b %x 2 + c 2^ 3 + d Xx* == %xy, 

 a tx 2 + b 2x z + c tx* + d 2^ = 2«fy, 



In order to simplify the calculations the experimental 

 results were reduced so as to apply exactly to whole numbers, 

 the rate of change being determined by means of the general 

 curve. The maximum correction which had to be applied in 

 any case was less than '05°, and any error in applying this 

 must have been quite inappreciable. The reduced values are 

 given in the lower part of Table I. 



The results of the examination are given in Table III. 

 When represented by two equations meeting at 16 per cent. 

 we get, as with the graphic method, a total error which is 

 practically identical with the ascertained experimental error 

 — *95 as compared with 1*0 (column in.) ; whereas an attempt 

 to represent the points by a single equation produces a result 

 with a total error 633 times greater than the experimental 

 error. Columns iv. to ix. show, moreover, that any attempt 

 to alter the position of the change of curvature to either side 

 of 16 per cent., induces a large increase in the total error, 

 and makes this error far too big for the drawing to be con- 

 sidered acceptable : also the farther it is shifted the larger 

 does the error become. The last three columns in the table 



* It may be convenient to quote the following equations : — 

 1+2+3 ... + „ = !*£!>, 



P+ 2' + 33.,. + ^ ra(ra+1 f" +1) , 



1*4: 3*4. V i ■.... n(n+l)(2n+l)(3n*+Sn-l) 

 1 +J +6 . . .+» - ^ , 



1.+S-4*. . .+„» = g(»+l)W+2»-l) ) 



1«_L96 . 3 6 . n e = »(»+l)(2»i+l)(3w 1 +6n 3 -3n+l). 

 "*" T 42 



t Mr. Hayes has suggested a method by which, when the values of x 

 differ by unity, the calculation of the constants may often be considerably 

 simplified. If the number of points is odd, we mav take the middle 

 point as origin, which gives -2, -1, +1, +2, &c. as "the values for*/ of 

 the other points, and the normal equations then become 

 na +2c'Sx 2 = 2y, 

 2& 2 x*+2d ^x i = ^xy, 

 2a2x 2 +2c2 ir 4 =2* 2 y, 



26 2.r 4 4-2rZ2x 6 = 2ary 

 The method can also be applied with slight modifications in cases 

 where the number of points is even, or where the constant difference 

 between successive values of x is not unitv. 



