Elements of analytical geometry and the differential and integral calculus . equired to find il^ and B^, which together should equalA^-^B^, or 136, and the area NCPM, which should equal AB, or 60, if theforegoing theory is true. Equation (5) will give us the value of n as follows : 100|+36|-^3=0. Or : 36^3 100 Log. 36-f-|-log. 100. Plus 10 added to the index to correspondwith the tables, gives for the log. tangent of the angle n, whichgives 31° 56 42, and the sign being negative, shows that 31° 56 42 mustbe taken below the axis of X, or we must take the supplement of
Elements of analytical geometry and the differential and integral calculus . equired to find il^ and B^, which together should equalA^-^B^, or 136, and the area NCPM, which should equal AB, or 60, if theforegoing theory is true. Equation (5) will give us the value of n as follows : 100|+36|-^3=0. Or : 36^3 100 Log. 36-f-|-log. 100. Plus 10 added to the index to correspondwith the tables, gives for the log. tangent of the angle n, whichgives 31° 56 42, and the sign being negative, shows that 31° 56 42 mustbe taken below the axis of X, or we must take the supplement of it, 7T, whence n=148° 3 18, and (n—m)=118o 3 18. To find A^ and B*^, we take the formulas from Proposition ,2__ A^B __ 100-36 _3600. •30-1-52 ioo-t+36-f 52 ^/2_ A^B^ 3600 -4^8in.^31°6642+(31°6642)~27-99+26^ 62 ANALYTICAL GEOMETRY. This agrees with scholium 1,As radius Is to ^^() So is sine {n—m) 61° 56 42 log. MK— Log. B=^ log. () ^^=60. log. 60= PROPOSITION find the general polar equation of an ellipse. If we designate the co-ordinates ofthe pole P, by a and b, and estimatethe angles v from the line PX par-allel to the transverse axis, we shallhave the following formulas : x=a-\-r ;. y=5-f-r sin. v. These values of x and y substituted in the general equation will produce ^2 sin.^?; T^^^A^ 1 r+Ah^-\-B^a=zA^B\ JS^cos.^v -^-^B^aco^.v]for the general polar equation of the ellipse. Scholium 1. When P is at the center, a=0, and 5=0, andthen the general polar equation reduces to ^2^ A^B^ A^&m.^v+B^QO^.^va result corresponding to equations (P) and ( Q) in Prop. IX. Scholium 2. When P is on the curve A^h^+B^a^zzzA^B^,therefore A^^m.^vB^cos.^v r=:0. -^^B^] This equation will give two values of r, one of them is 0, asit should be. The other value will correspond to a chord, THE ELLIPSE. according to the values assigned to «, h, and v,la
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