168 Suggestions /or iimplifijing 



In this class of persons I must observe tliat I feel myself 

 included ; for it was not till I had given Mr. Ivory's paper a 

 very attentive consideration, and had been at considerable 

 pains in arranging the calculations of an example, that I fully 

 perceived the great practical importance of this result of his 

 able and ingenious view of the problem. 



The expressions to which I have alluded as admitting of a 

 convenient transformation are these ; viz. 



^, , sin A + sin A' „ -, ,. -r> sin A— sin A' „ i 7 j 



« A = and "B = ; where h and 



Ji are the altitudes, and the logarithms of A and B are the 

 quantities required. 



^, sin A + sin A' .' h + h! h—h' 



JSow, ^ = sm — ^ - , cos —^- ; 



, sin /(— sin/j' . h — /*' h + h' . « 



and = sm — — . cos — - — ; expressions trom 



which the logarithms of A and B may be obtained at once. 



With these alterations in the forms of the expressions for 

 A and B, Mr. Ivoiy's formulas for computing the latitude 

 may be arranged as under ; / being the half interval of time 

 between the observations, h and h' the altitudes, D the decli- 

 nation, and L the latitude. 



1. Sin a = cos D. sin t. 



2. Cos b = sin D. sect a. 



„ o- h + h . h — h' 



3. om c = cos — — -. sm — — . cosect a. 



4. Cos d = sin — ;^ — . cos —^ — . sect c. sect a. 



5. Sin L = cos c. cos {h + d). 



It may be observed on these fornmlas, that a, sect a, (which 

 is wanted twice,) and cosect a, are obtained at the same opening 

 of the table ; and so are cos D and sin D ; sin and cos of 



: sin and cos of ; and c, cos c, and sect c; so that the 



number of openings of the book of tables is comparatively small. 



The demonstrations of the formulas may be seen in Mr. 

 Ivory's paper, which also contains the investigation of a very 

 neat method of estimating the effect of the change of the sun's 

 declination, in the interval between the observations, when it 

 is deemed of any consequence to advert to it. The practical 

 rule from the above formulas may be thus given in words, ob- 

 serving that in evei-y case the tens are to be rejected from the 

 indices of the logarithms. 



Practical Mule: 



1. Add together the sine of half the elapsed time, and the 



