268 On the Systematic Fitting of Curves 
2/ = (/> (0) + (0) + ^2 0" (0) + r (0) + ... 
= Ho + + as + as Y^^^ + s^y- 
Here «„, a^, ol^, ... etc. will be functions of the n constants Cj, c.2, ... c„ of the curve. 
Hence theoretically we can find the n c's in terms of Oo, a^, a.,, ... «n-i- We should 
thus be able on substitution to express a„, a„+], ... etc. in terms of a,,, aj, ... ofn-i. 
Now consider the first n a's as the constants of our curve and it will be expressible 
in the form : 
of x"^^ 
1 . z \n — I 
+ (tto, Wj, Ka, ... a„_i) |— 
[TO 
+ 0"+H«o, oil, Ha, ••• a«-i) 
a; 
.n+\ 
ft + 1 
+ etc (i). 
Next let y' be the ordinate corresponding to x given by observation, then y — y' 
will be the distance between the theoretical and observed curves at the point 
corresponding to x, and our object is to make the values of this as small as possible 
by a proper choice of a^, oli, a.^, ... a„_i. This may be done by the method of least 
squares or making 
Jiy — y'f dx = a minimum. 
This obviously gives a very good method, if not the " best," a term incapable of 
definition. The resulting equation, since y is the variable, is 
I(y-y') Bydx = 0 (ii). 
Now By depends on the variation of Mq, ... a,j-i, or 
X" 
+ etc. 
= 8ao 1 + +-^r— Y + ... 
\ dua \n aoTo |»i + 1 
+ .. 
1 . 2 aa^ \n da.^ \ n + 1 
+ 
