Adjustment of Observations. 179 



and yet it may be found that the arithmetic means of the 

 measurements made on each angle do not add up to 180°. 



This problem is of great importance in some of the 

 practical applications of science, such as surveying It is 

 not of much importance in pure physics, for we very 

 seldom require to know with great accuracy and certainty 

 the value of any directly measured magnitude; it is 

 derived magnitudes that are important*. 



(3) Measurements have been made on many sets of 

 magnitudes (x, y, z, . . .), which are known to be re- 

 lated by a numerical law of which the equation is 

 /(#, y, z, . . . a, b, c, . . .) =0, the form of the function f 

 being known, but not the values of the constants a, b, c, . . . 

 For example, measurements have been made of the activity 

 of: a pure radioactive substance (I) at various times (t). It 

 is known that I and t are related by the equation I = I .e~ M . 

 It is found that no values can be assigned to the constants 

 which are such that all the measured sets actually satisfy 

 the equation. It is required to determine those values of 

 the constants which are to be regarded as the true values. 



This third problem is of immense physical importance, 

 and the solution or" it is involved in almost every expe- 

 rimental research. It is often solved by graphical methods : 

 numerical computation is used only when the number of 

 constants is too great to be represented on a plane diagram, 

 or when it appears that graphical methods do not utilize 

 fully the accuracy of the observations. But it is desirable 

 to discuss methods of computation applicable even to those 

 cases where graphical methods can also be used ; for it will 

 be admitted that both methods should be founded on the 

 same principles. 



3. The principle of solution. 



The accepted method o£ solving the second and third 

 problems, which is embodied in the Method of Least Squares, 

 depend^ on the assumption that the true values of the mea- 

 sured magnitudes in the second problem or of the constants 

 in the thud are those which make the sum of the squares of 

 the residuals a minimum : the residuals are the differences 

 between the measured magnitudes and those calculated from 



* The reason is that pure science is not concerned with the investigation 

 of the properties of individual objects, but only with the establishment of 

 laws. A magnitude which is determined by a law, and therefore important 

 for pure science, is always a derived magnitude. 



N2 



