96 PHILOSOPHICAL TRANSACTIONS. [ANNO 1797. 



are applied to practice. To this paper is added another new method for the same 



purpose as this 2d part, by Mr. Cavendish, as follows. 



Extract of a Letter from H. Cavendish, Esq. to Mr. Mendoza y Rios, Jan. 1795. 



" The methods in which the whole distance of the moon and star is computed, 

 particularly yours, require fewer operations than those in which the difference of 

 the true and apparent places is found; but yet, as in the former methods, it is 

 necessary either to take proportional parts, or to use very voluminous tables; I am 

 much inclined to prefer the latter. This induced me to try whether a convenient 

 method of the latter kind might not be deduced from the fundamental proposition 

 used in your paper, and I have obtained the following, which has the advantage of 

 requiring only short tables, and wanting only one proportional part to be taken, and 

 I think seems shorter than any of the kind I have met with. 



" Let h and h be the apparent and true altitude of the star; / and l the apparent 

 and true altitude of the moon, g and g the apparent and true distance of the moon 

 and star. Let the sine and cosine of g = d and $, the sine and cosine of / = a and 

 a, the sine and cosine of h = b and (3 ; and the sine of the actual and mean hori- 

 zontal parallax = p and tt ; and let the sine of l = a — m -+- pe 3 'and its cosine = 

 a (1 _|_ y. — p^ and let the sine of h = b — n, and its cosine = (3(1-1- v). 



" Then the cosine of g = i (l -f ^ — pi) (l + >) + (a — m + pe) (b — n) — 

 ah (1 + /a — pi) (1 + *)> which equals $ -f- fy, -f h — tyt + *fw ~ H*» + ab — 

 l m _[_ Ipe — an + nm — n P e — a ^ — •%• H" °bp £ — Q bv — abpv -J- abvpt = $ -f- 

 jp -L. $ v — $p £ — bm — bap + bpe + bapt — an — abv -f- nm — npe — abpv -|- 

 abvpt -\- fy*v — Sttw. 



" To make use of this rule, it must be considered that the quantity Spy — $p& 

 is so small that it may safely be disregarded; but nm — npe — abpv -f- abvpt, if the 

 altitudes are not more than 5°, may amount to about 12 /; , and therefore ought not 

 to be neglected. The quantity e + at also differs very little from J, but is not 

 quite equal to it. Let therefore a table be made under a double argument, namely, 

 the altitudes of the moon and star, giving the value of ... . nm — n-ne — abpv + 

 abvrrt + k*e -f- ba-Ki — bir, answering to different values of these altitudes, which 

 call A. Let a 2d table be made under a double argument, namely, the altitude of 

 the star and the apparent distance of the moon and star, giving the value of tr, which 

 call d. Let a 3d table be made with the observed altitude for argument, giving the 

 logarithm of am -\- a 2 /*; and let this quantity, answering to the moon's altitude, 

 be palled m, and that answering to the star's altitude, n ; observing that the same 

 table will do for the moon and star; but a 4th table should be made for the sun, so 

 as to include its parallax; and, lastly, let a 5th table be made, with the moon's al- 

 titude for argument, giving the logarithm of - = — , which call c. Then will cos. 



g = <J — Sapc £- -f bp + d — A H 



" It must be observed that Sapc = Spi — — , whereas it ought to equal Spt 





