Adjustment of Observations. 183 



arranged in increasing (or decreasing) order of y. If there 

 are N equations and n magnitudes (x, y, z, ... .), and if 

 N=/m-f q, where p, q are integers, then the first normal 

 equation is to be formed by adding the first p equations in 

 this order, the second by adding the second p equations, 

 and so on until n—q normal equations have been formed; 

 the other q normal equations are to be formed by adding p + 1 

 observational equations taken in order in the same way *. 



The basis of part of this rule is obvious. The method 

 provides that the number of observational equations added 

 to form a normal equation is (as nearly as possible) the 

 same for each normal equation. Since the assumption 

 underlying the method is only true if that number be large, 

 it is desirable to prevent it being smaller than it need be in 

 any one case ; that result is obtained by making the number 

 equal in each case. The basis of the remainder may be 

 seen by considering the case when there are only two 

 magnitudes, x, y. Then we are practically taking the 

 mean of each of two halves of the observations and deter- 

 mining b and m from these two means. The determination 

 might be made graphically : we might plot the points 

 representing the two means and draw a straight line 

 through them. It w T ould then be obvious that the accuracy 

 with which the straight line could be drawn would be 

 greater the greater the distance between the two points. 

 It is desirable therefore that the difference between the 

 two means should be as great as possible : this condition 

 is obtained by arranging the observations in the order 

 proposed for the purpose of forming the normal equations ; 

 for one mean is that of all the smallest values of y and 

 of all the smallest (or largest) values of x, whereas the 

 other is that of all the largest values of y and all the largest 

 (or smallest) values of x. 



4. Errors. 



It will be convenient also to express the matter ana- 

 lytically. The normal equations will be 



X 1 = bY l + cZ 1 + .. 

 X 2 = 6Y 2 + cZ 2 + . . 



m > ' (3) 



J 



* It may be observed that if q is not the result obtained will be 

 slightly different according as an increasing or decreasing order of y is 

 adopted. But the differences arising from this latitude of choice are 

 completely negligible. 



