[ 513 j 



LXI. On a Method of Finding a Parabolic Equation of the 

 vth Degree for any Graphically Faired Curve. By T. 0. 

 To bix, M.A. 



IF a mean curve be drawn through a series of points 

 plotted with respect to a set of rectangular axes, then 

 an approximate equation to the curve may be found in the 

 form 



t/ = a + a l .c+ . . . +a r .c>' 

 by the following method, which lends itself readily to 

 arithmetical computation. 



Take " n " equidistant abscissae, such that 



a?=l, 2, 3 . . . n correspond to 

 y=yi- y 2 , y% - ' • l)n as measured from the curve. 

 Then the equations of condition to determine the constants 

 « , a i? a 2 ? & c *> are 



j/i = «o + % + a 2 + . . . + a r , 

 y 2 =a Q + 2 . «i + 2 2 . a 2 + . . . -t-2 ; '.a,., 



y,, = a + >i . rtj + n 2 . a 2 + • • • + ra r • «r- 

 These give 



i — nCtfi + „C 2 j/ 2 • • • ( — Ty n 



-^[i ..d-2. „a 2 + ...c-)^- 1 ^ ] 



-a 2 [l 2 . n C 1 -2' 2 .nC 2 + ...(-)- 1 ,n 2 ] 



that is, 1— „Piyi+nC 2 y2"'(— ) n y»=l— «o? • • O) 



since w**— ,A(n— l) r + . . . (— ) n - 1 l 7 ' n 1 = so long as^> r. 

 This latter relation may be easily verified by considering 

 the identity 



111 1 __ 1 % n (~) n ° 8 



y ' y -+- 1 ' y + 2 ' ' y + n n\ g=0 y + s ' 



which may be written 



=;|(-),r.[i + 2 ( -).(;)']. 



Since there is no power of - on the left-hand side less 



y 



* Communicated bv the Author. 



