706 PHILOSOPHICAL TRANSACTIONS. [aNNO ]7Q6. 



formulas: his sole end being to show how to obtain these by the elements. 

 He gives, for example of their utility, the facility with which we obtain the 

 differential logarithmic equation; from which reciprocally the calculation of 

 logarithms is deduced. 



Since «- = 1 + A2 + ^ z' .+ — *^ z' + &c. 

 -^— = A (1 + Az + y— 2 z- + YTTTs 2 + &c. = A . a^ 



, d'.ir T , d' a' , , d* a' . 



hence -j— = a- az; and -— - = a^ a'; and —^r = A* . a''. 



(tz az* (Iz 



Reciprocally. Since a log. 1 -\- v = v — - v^ -\- - v^ — &c. 



d . log. I + V , ,2 3 , o 1 



dv ' I + V 



§ 8. In the 2d Part, on the sines, cosines, and tangents, of circular arcs. 

 Mr. L'H. commences with 2 well-known lemmas: viz. 1. The difference of the 

 sines of two arcs, is equal to double the product of the cosine of their half 

 sum by the sine of their half difference. 2. The difference of the cosines of 

 two arcs is equal to double the product of the sine of their half sum and of 



their half difference. That is, sin. a — sin. b = 2 cos. ° ■ sin. °' ~ : 



2 2' 



and cos. b — cos. a = 2 sin. sm. , 



2 2 



^ Q. Let sin. a, sin.. 2a, sin. 3a, sin. Aa, &c. be sines of arcs in arithmetic 

 progression increasing, for ex. as the natural numbers. And let there be taken 

 the differences of the successive orders of these sines; by which we obtain the 

 1st, Diffs. — 2 sin. \ a (cos. f a, cos. a a, cos. \ a, cos. -§- a, &c.) 

 2d, Diffs. — 2^ sin.^ -i- a (sin. 2a, sin. 3a, sin. Aa, sin. oa, &c.) 

 3d, Diffs. — 2' sin.^ i a (cos. f a, cos. ^ a, cos. f a, cos. y a, &c. 

 4th, Diffs. + 2^ sin.'' -i- a (sin. 3a, sin. Aa, sin. 5a, sin. 6a, &c.) 

 And in general, the 2m diffs. + 2" sin.-'" ^ a (sin. m + 1 • a, sin. m -\- 2 .a 

 sin. m -{- 3 . a, &c. 



2m + 1 diffs. ± 2^'" + ■ sin.^" +'\a (cos. ?^^ a, cos. ?^^^ a, cos. "^-^-I 

 a, &c.) 



§ 10. In like manner, let cos. a, cos. 2a, cos. 3a, cos. Aa, &c. be the co- 

 sines of arcs in arith. progression, increasing for ex. as the natural numbers. 

 And let there be taken the diffs. of the successive orders of these cosines; by 

 which will be obtained the several orders of differences, thus: 



1st, Diffs. — 2 sin. -i- a (sin ^ a, sin. a a, sin. -^ a, sin. |- a, &c.) 



2d, Diffs. — 2'^ sin.- i a (cos. 2a, cos. 3a, cos. Aa, cos. 5a, &c.) 



3d, Diffs. + 2* sin.^ -i- a (sin. f a, sin. -|- a, sin. :J- a, sin. y «j &c. 



4th, Diffs. + 2' sin.'* J- a (cos. 3a, cos. 4a, cos. 5a, cos. 6a, he. 



And in general, the 2m diffs. + 2"" sin.*-" \ a (cos. /w + 1 . a, cos. ?rt + 2 . a, 

 cos m + 3 . a, &c. 



