94 EIGHTH REPORT 1838. 



If we retain only the terms of this equation in which x and y 

 are of the first dimension, we have the equation (2) already 

 obtained. 



To advance another step in the approximation, we should in- 

 clude the terms in which x and y are of the second dimension ; 

 and we shall thus have six unknown coefficients L, M, N, P, Q, 

 R, to be determined. For this purpose, the equations (in number 

 the same as the stations of observation) are to be combined by 

 the method of least squares ; and the six resulting equations 

 will give, by elimination, the quantities sought. 



The coefficients L, M, N, &c. being known, the Hne oi given 

 dip is 



R y^ + Q a: y + P a;2 + N y + M ^ = K, (8) 



in which K denotes, as before, the particular value of ^ — {z). 

 Here, then, the isoclinal hne is of the second order ; and its 

 species is determined by the relation of the first three coeffi- 

 cients, P, Q, R. The equation of the curve being found, it is 

 easy to construct it graphically by points. 



The preceding solution of the problem is probably suffi- 

 cient for all purposes ; but the determination of six unknown 

 quantities by the method of least squares, when the equations 

 of condition are numerous, is a formidable labour ; and it is 

 therefore important to consider whether we can safely stop 

 short at any step of less generality. Now it is easily seen that 

 in most cases to which we have to apply this method, the iso- 

 clinal line may be represented by the equation 



p ^2 ^ N y + M ^ = K, (9) 



in which there are only four coefficients to be determined*. 

 This equation (considered as belonging to a plane curve) is that 

 of a parabola. 



The equation, being linear m. one of the co-ordinates, is very 

 easily constructed by points. 



* This is evident from geometrical considerations. 



Let L M be a portion of the cm-ve, re- 

 ferred to the axes of co-ordinates O P, O L ; 

 and let L Q be its tangent at the point L, 

 making with the axis of abscissiE an angle 

 whose tangent is u. The ordinate of the 

 curve P M, is equal to P Q + Q M. But 

 P Q, the ordinate of the tangent, is equal to 

 ax + h,b denoting the ordinate at the ori- 

 gin, O L. And the sagitta Q M, is pro- 

 portional to Q L^ the arc being small in 

 proportion to the radius of curvature ; i. e. 

 Q M = A X Q L2 = /f (1 -i- a2) .x2 = ca!^ 



