172 PROCEEDINGS OP THE AMERICAN ACADEMY 



a; is a simple function ; but if x is a complex function the solution will 

 no longer give the true values of the separate unknown quantities, 

 though it may yield such values as will give the most probable sum of 

 A a, Bb, G c, Dd, &c, with respect to their effect upon x. 



Let us take, as an illustration, the ordinary equation for the reduc- 

 tion of transit observations. The fundamental equation may be put 

 under the following variety of forms : — 



(a) Q = &T+[T-K.A.]+Aa 



(b) =&T+lr-R.A. : ]-\-Aa-! r Bb 



(c) =AT + [T-R.A. : ]+Aa-\-Bb + Cc 



(d) =AT + [T -R.A. : ]-\-rh-\-Aa + Bb-\-Cc 



(e) =AT + lT -R.AJ] + Th-{-Aa-\-Bb + Cc-\-Dd 



(/) = AT + [r -R.A. ]-f^e-f T h + Aa+ Bb+Cc -\-Dd 

 (g) =AT + [T -R.A. : ]-\-Be + th + Aa + Bb + Cc+Dd-\-C 



If the level b and the collimation c are obtained independently of 

 the observations by direct measures, then, neglecting the small terms 

 which follow, for any time, T, and with the given right ascension, 

 R. A., the only unknown quantities in equation (a) are the clock 

 error AT and the azimuth term A a. A solution of a series of 

 equations of this form will give the most probable individual values 

 of a and AT. 



If the level term B b is unknown, the general equation takes the 

 form (b). Notwithstanding the fact that the equation is somewhat 

 more complex in its structure, the solution by least squares will give 

 the most probable individual values of a and b, if the stars are selected 

 with reference to a proper distribution of positive and negative values 

 for A and B. 



If the collimation term C c is unknown, the equation takes the form 

 (c). Here a solution by least squares will not give the most probable 

 individual values of a, b, and c, unless the observations are arranged 

 with proper reference both to the magnitude and the sign of A, B, 

 and G. Even when these precautions are observed, the value of c from 

 the solution will rarely agree exactly with the value obtained from 

 reversal or from collimators. 



If, for any star, the observed time T is written T Q -\- r h, the term 

 T h being the hourly rate of the clock multiplied by the interval t 

 between T and T , the equation takes the form (d). We now intro- 

 duce an unknown quantity depending on another instrument, viz. the 

 clock. 



