XCvi THEORY OF ERRORS. 



This equation includes all cases. Its analogy with (a) should be noted, since 

 the step from (a) to {l>) is clear when the correct form of (a) is known. Mistakes 

 in the application of (^) are most likely to arise from a lack of knowledge of the 

 independaitly obscnxd quantities x, y, z, . . . or from a lack of knowledge of the 

 true form of (a). Hence,* in deriving probable errors of functions of observed 

 quantities attention should be given first to the construction of the expression for 

 the actual error (a). 



A few examples may serve to illustrate the use of (a) and (d). 



(i.) Suppose 



Then 



(2.) Suppose 

 Then 



9F dV ^ 9V ^ . 



A F = ae^, + (^^ — a)e,j + (^ + c)e^, 



e"' = a^e/ + (^ - aye,' + (/. + cy-^^ 



(3.) Suppose 

 Then 



9V_ _ a 9V_ b_ 9V__ _ -zhy 



9x ~ x'' dy ~~ / 9z 2« ' 



A 7^ CI , b 2 by 



^^ x''''+?'"~^'" 



F= a; log X -f- ^' sin y -\- c log tan z. 



9V_ Ojx t 



9x X 



and 



^^ 



(^)' 



(4.) Suppose the case of a single triangle all of whose angles are observed. 

 What is the mean error, ist, of an observed angle; 2d, of the correction to an 

 observed angle ; and 3d, of the corrected or adjusted angle ? 



Let X, y, z denote the observed angles, /, ^, r their weights, and Ax, Ay, Az 

 the corresponding corrections. 



Then, as shown on p. Ixxxvii, 



Ax -\- Ay -\- Az =z c = 180° + sph. excess — (x + J' + -) 



= error of closure of triangle, 



Q 



Ax=^, Ay = 2, Az=Q^. 



p q ^ 



* As remarked by Sir George Airy in his Theory of Errors. 

 t M = modulus of common logarithms. 



