8 PHILOSOPHICAL TRANSACTIONS. [aNNO 1785. 



will give the difference in declination of that star from the angle where those 

 obliques meet, as the whole interval does in the rhombus. 



But it may perhaps be of service to astronomy, or at least not unacceptable to 

 those gentlemen who use the rhombus, says Mr. W., that I should subjoin 

 another formula (contrived for me the last summer by my son, now Mathema- 

 tical Lecturer at Sidney College, Cambridge), for investigating the comparative 

 right ascensions and declinations of stars observed by it, when the instrument is 

 not placed truly in the plane of the equator. I was led to wish for some such 

 formula, in consequence of an ingenious paper, kindly communicated to me by Sir 

 H. C. Englefield, Bart., p. r s., giving an account of his method of doing it by a 

 scale and figure ; which, though very easy when one is provided with such a 

 scale, appeared to be of less general use than by calculation ; and I do not know 

 that any thing of the kind is to be met with in any publication. 



Let the angle dll, fig. 3, pi. 1, which by construction is 63° 26', be called a ; 

 the diagonal ll be called b \ 



The larger interval observed between the passage of a star by an oblique and 

 the horary wire, as hc^ be called m ; 



The smaller ditto of the same star, as cd, be n ; 



The large ditto of another star, as (Sy, be ^ ; 



The smaller ditto, as y^, be y : 



Then— ^"^^^ = tangent of the angle which ll makes with a parallel of decli- 

 nation : call this a : the angle q being thus found, then ^^" "^' ^ sin.(a+g) ^ 

 ^ 030 ^ j^^ sm, a 



q. = difference in declination between the two points on the vertical wire where 

 those stars pass it. Which, being in time, must be converted into degrees, and 

 multiplied by cosine of declination as usual, to give the true difference in decli- 

 nation between the stars. 



A J ^u • • 2 (•» co») X sin. (a + q) ^^ • ., i-ar 



And the same expression, viz. — ^ : — -^^ ^^ X sin. o = the difference 



^ R X sin. a ^ 



in a. r. between those two points ; to be applied as a correction to the observed 

 times. 



The same may be done by the larger intervals m and [/., only by substituting 

 a — a instead of a 4- q, thus: '^ '^ xsm.^a—q ^^^ __ ^^ff^^^^^^ -^^ jg_ 



^ ' ' ' R X sin. a ^ 



clination as above; or X sin. q = ascensional difference. 



If the stars differ too much in declination to come within the expression above 

 (as N** 1 and 3) then the differences of the angles d and e in declination and 

 right ascension may be found thus : 

 — : — ^ ^"^' ^ = diff. in declin. between d and e ; ' — ^1_!!!L? = their ascen. diff., and 



R R ' 



the difference of each star from its respectively nearest angle of the rhombus. 



