378 Reply to Mr. Ivory's Remarks, &c. 



gent '41201. We shall also have, for the ordinate, y = \- 



1-05635 bx' + 2-807 h^ x^ + 19 b' x'' = -004160 + W465 

 + 001071 + -000057 + [-000003] = -032756, with very lit- 

 tle uncertainty, even with regard to the last place of decimals. 



We may now proceed to compute the remaining increment of 

 the sine, by the method laid down in the Eleinentary ILlustra- 

 tmis, p. 95. The original equation being hex&fyxAx = msx, 

 or qj'yxdx=sx, we have gyxdx = sdx + xAs, (]y = 



— 4-4^, and — ^ = o w — — = A, for the first coefficient of 

 AX', then d-^^ =5^2/ -+ -^5 and, if t be the tangent of 



..... du , . , , d-s , ds , s 



the Hichnation, -~- beme equal to /, -^-- ■= q t 3 — r "" — 



2 X XX i X XX XX ^ \ X / XX 



Again, since d f = --, if ti be the cosine of the inclination. 



, d^s nds dA Ada- ds 2sdx , A^s ^ a A 



d -j-; = ^^ 1 T-J a"d TT = C = 3 



ax- ?f* X txx XX x^ ax^ u' 



B.A A 2s »/9 2\ B 2s, ^, 



+- ,- =A(-^ +— ) r. Lastly 



X XX XX i3 \ u> XX / X X' 



for D, since dM = — id^, and -^ ~ r- = — tA, we obtain 



' do; ax 



1 d'« J A / 7 . 2 \ . /Sqdu 4dx \ dB Bdr 2ds 



d — = d A( -7 + — ) - A( -^ + -^3^ ) H -T- 



dx^ \ m' XX y \ u* x^ / X XX x* 



i-i ' dx« \ w XX / \ u^ ar3 / x xx 



a^ I* M* l3 \ \l? XX / X l' 



thus obtain the series A5 = '25139 + -08548 + '02314 + 

 •00655 + [-00218], making the remainder i- of the last term, 

 since the ratio of the preceding terms differs little from \ ; so 

 that we have 5 = '74960; which is less than '75 by '0004, or 

 about stt'o-o of th^ whole only, without a possibility of any error 

 in the last place of the depression '00416, if the numerical com- 

 putation is correct; unless indeed the defect of the series should 

 still have made even this value a little too great. 



LXVir. No- 



