514 Mr. T. C. Tobin on a Parabolic Equation of the 



than the nth, the coefficient of — on the right-hand side 



will be zero if n > r, and this coefficient is 



W f_ ji 1 (n-1)-+ fc ..(-)- 1 l-. w C 1 . 



Hence, instead of equation (1), we may write 



1 -d =l- r+1 C 1 y 1 + r+1 C 2 y 2 . . . (— )' +1 y r +i. . (2) 



Having thus determined a , the original equation for the 

 curve may be written 



II ri 



7] = = a x + a 2 x + • . . + a r x r ~ l ; 



x 



and from this we obtain, in a similar manner to the above, 



1 — a Y = 1 — ;Ci . Vl + rC 2 . ^ 2 • • • ( — )'V • • (#) 



Proceeding as before, we may now write the equation for 

 the curve in the form 



v' = — - l = a 2 + a z x + • • • + «»• • «v r_2 , 

 and obtain 



i-^=i- r _ 1 c 1 .v+ ...(-y- 1 .^.!, . (4> 



and so on. 



If the curve cuts the axis of y so that ijo = a is known y 

 then a x may be determined as before from 



7) = l — -2 = «i -f a 2 # 4- ... + a r . af"" 1 . 

 a; 



To illustrate the arithmetical convenience of the method,. 



let the observed "y" values corresponding to equidistant 



abscissae be as follows : — 



y =l-00==Oo' 



yi = l-14 



7/2=1*36 



2/3 = 1-70 

 3/4-2-11 

 the assumed equation of the curve being 



y = a + a\X + a 2 .r 2 + a 3 <Z' 3 -f a 4 .£ 4 . 



The calculation of the coefficients can be arranged in 

 tabular form (p. 515). 



The method may, of course, be used to obtain an approxi- 

 mate equation to any arc of a continuous curve of one signed 

 curvature whose ordinates can be measured at equidistant 

 abscissae values. 



