94 ON THE FORMULA FOR CALCULATING 



undergoing the toil and loss of time which would attend a formal recom- 

 putation of the whole set of observations, and it will be remarked that 

 the second hypothesis will also enable him to compute, by means of 

 ditFerences, a series for many nights in succession with quite as much 

 correctness as if entire quantities had been used ; for in that case he 

 has only to calculate the elements for the first night, and substitute for 

 the value of d 6 the increment or decrement of polar distance, the other 

 terms being virtually constant. 



It has always, however, been an evil complained of in operations of 

 this kind, that by limiting the case to the actual time of maximum 

 Azimuth, the powers of the observer are much curtailed, because he can- 

 not take more than one observation on the same night. 



If observations, taken intermediately between the culmination and 

 the time of the greatest Azimuth, were to be computed with reference to 

 the meridian, it would be indispensable to employ large quantities, and 

 the operose formulae of Spherical Trigonometry, which would not only be 

 laborious, but would not give so much accuracy as the method of eliciting 

 the correction by means of differential terms. 



z 



I shall explain this better by a reference to 

 the diagram in the margin, wherein the two Arcs 

 ZS, ZS' are drawn very near to each other, the 

 angle PSZ being a right angle, and it will be seen 

 immediately how much more easily and accurately 

 the angle PZS' may be found by computing the 

 partial angle SZS', and deducting it from PZS, 

 than by direct computation of the entire angle 

 PZS' itself. ^ 



I have introduced, in the first part, the ordinary rules for com- 

 puting the elements at the time of maximum, with the view that 

 those, for whose use these formula are intended, may not need a 



