Elements of analytical geometry and the differential and integral calculus . anged at the same time inreference to the oblique system, we shall have — , ^ , yco^.m—x^ X =za-\-——-j~—r— y =H sin.(w—m) sin. (72—m) We close this subject by the following EXAMPLE. The equation of a line referred to rectangular co-ordinates is y=:axA-y. Change it to a system of oblique co-ordinates having the sam®zero point. Substituting for x and y their values as above, we have a;sin. m-\-y sin. n=^a{x cos. w^-|-ycos. n)-\-b\Reducing , (—)a;, b sm. n—a cos. m sm. n—a cos. m 26 ANALYTICAL G
Elements of analytical geometry and the differential and integral calculus . anged at the same time inreference to the oblique system, we shall have — , ^ , yco^.m—x^ X =za-\-——-j~—r— y =H sin.(w—m) sin. (72—m) We close this subject by the following EXAMPLE. The equation of a line referred to rectangular co-ordinates is y=:axA-y. Change it to a system of oblique co-ordinates having the sam®zero point. Substituting for x and y their values as above, we have a;sin. m-\-y sin. n=^a{x cos. w^-|-ycos. n)-\-b\Reducing , (—)a;, b sm. n—a cos. m sm. n—a cos. m 26 ANALYTICAL GEOMETRY. Polar, €o-ordioates. When a ]ine is conceived to revolve round a point, that pointis called a fole, and any other point in such a line referred toco-ordinates, is denominated the system of polar co-ordinates. Conceive the line AB to revolveround the point A a-B a pole. LetAB=r. It may be a variable dis-tance, and it is then called the raditisvector. Put the variable angle BAD^v,AD=x, DB=y, then by trigo-nometry x=r COS. V, and y=r sin. Now from the first of these we have (v may re- volve all the way round the pole), and as x and are bothpositive and both negative at the same time, that is, both posi-tive in the first and fourth quadrants, and both negative in thesecond and third quadrants, therefore r will always be positive. Consequently, should a negative radius appear in any equation,we mtist infer that some incompatible conditions have been ad-mitted into the equation. Scholium 1. If we change the origin now from A to A,writing x and y for the corresponding co-ordinates, we shallhave x=a-\-r COS. V y=b-^r sin. V. Scholium 2. If in place of estimating the variable anglefrom the line AD the axis, we estimate it from the line Alfwhich makes with the axis the given angle £[AI)=^m, we shallhave x=a-{-rG08.(v-\-m). y=b-\-r sin. (v-\~m). THE CIRCLE. 27 CHAPTER of the second order. Straight lines can be represented by equ
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