. A new treatise on the elements of the differential and integral calculus . ~ dx dt dt and we may have or or -/1(S)*+(S)+(S)T- 254,, To convert the formulae of the last article into polarformulae, take the pole at the origin of co-ordinates, and denoteby d the angle that the radius vector makes with the axis of2, and by g) the angle that its projection on the plane (cc, y)makes with the axis of x; then we have the relationsx^^r sin. 6 cos. (p, y ^r sin. d sin. gy, s = r cos. 6. These three equations, together with the two equations ofthe curve, make five between which we may conceive r and qo
. A new treatise on the elements of the differential and integral calculus . ~ dx dt dt and we may have or or -/1(S)*+(S)+(S)T- 254,, To convert the formulae of the last article into polarformulae, take the pole at the origin of co-ordinates, and denoteby d the angle that the radius vector makes with the axis of2, and by g) the angle that its projection on the plane (cc, y)makes with the axis of x; then we have the relationsx^^r sin. 6 cos. (p, y ^r sin. d sin. gy, s = r cos. 6. These three equations, together with the two equations ofthe curve, make five between which we may conceive r and qoto be eliminated, leaving three equations between cc, ?/, 2, and6 : hence, x, y^ and z may be regarded as known functions of d. Therefore d nr ci ^ Cm CO —- r= sin. d COS. w r sin. 6 sin. o) —- + r cos. d cos. (p do ^ dd ^ dd^ ^ dy . dr . dcp -— sm. 0 sm. CD —- -\-r sm. 6 cos. qi -^ 4- r cos. 6 sin. cp, do ^ dd^ ^ dd^ ^ dz dr —- =: cos. d — r sm. 6: dd dd POLAR FORMULAE. 437 dxV , /dyV fdzV ^J + W+W ^ d(f,Vdo) = (|\V,.,i„. and -/)■+© -|- r^ sin.^ \ddjdo. do which, by changing the independent variable, may become /rlR\ 2 /rl^\ 2 ^ 1 or =/■©+ -y- I -f- r^ sm. (dcf/
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