262 Theory of Transit Corrections. 



emplified with reference to c, and consists in subtracting two 

 equations, one of which is the sum of several such equations of 

 the error of the clock as contain large coefficients of c, and the 

 other the sum of an equal number of such equations as contain 

 small coefficients of c, and such coefficients of a that their sum 

 will nearly balance and cancel the other coefficients of a in the 

 subtraction. Then the error of the clock disappears, the term 

 into which a enters being very small, may be neglected, and c 

 becomes known. The superior accuracy of this method, which 

 we find assumed, it may be well to establish. 



Before proceeding farther, however, it should be observed that 

 while the expression for the effect of the error of collimation 

 upon the time, needs in strictness of theory to be subjected to 

 the same modification as that which commonly expresses the time 

 of a star between two wires, the difference between the value of 

 this expression and the true correction, is always practically inap- 

 preciable. The time occupied by a star in passing from a small 

 circle of the sphere parallel to the meridian, to the meridian, is 

 equal to the space passed over, expressed in time, into the secant 

 of the declination ; but this space or the arc of the diurnal circle 

 intercepted between the two circles, is different for different de- 

 clinations, and hence the constant c can only represent it approx- 

 imately. In practice, however, the difference is of no conse- 

 quence. Let 0"05 be the greatest allowable error in the value 

 of the correction arising from the use of the form c . sec 8. Then 

 at 88° 30', the declination of Polaris, the error of collimation 

 must be 10 5 ; which is greater than it will ever be when any 

 sort of care is used in making the adjustment. 



The value of c as found by the common method, is much less 

 affected by errors of observation than that found from any three 

 equations. The effect of these errors on the value of c as de- 

 termined by the common method, is expressed by the algebraic 

 difference of two sets of errors into a coefficient which is a very 

 small fraction, and which may be made of any degree of sniall- 

 ness, while the effect of the errors of observation on c as found ^ 

 by the other method, is expressed by two terms each of which is 

 the algebraic difference of two single errors into a coefficient 

 which is probably integral. Now, since there is no probability 

 that the sum of these two errors is less than the greatest, if l£ 

 can be shown that the coefficient of either is integral, and that 

 there is no likelihood that its factor is any less than the corres- 

 ponding factor of the term which expresses the effect of the 

 errors of observation in the common method, it follows that this 



method. 



much 



Letx^JE —y -a.m -c.n 



#— iE 7 —y f —a.m f —c.n' 



