An elementary course of infinitesimal calculus . Fig. 112. When the lower sign in (12) is taken, the family includes the wrch rlt = \i.\i^ (13). See Fig. 112. 149] SPECIAL CURVES. 389 The ovals of Cassini are defined byrr =:T^ Ex. 3. ? (14). h being a given constant. Since for a point P in SS thegreatest value of rr is <?, it follows that the curve will consist oftwo detached ovals surrounding S, S, respectively, or of a singleoval embracing both points, according as I; < Fig. 113. In the critical case of h = c the curve is known as the lemniscate of Bernoulli; this presents itself i
An elementary course of infinitesimal calculus . Fig. 112. When the lower sign in (12) is taken, the family includes the wrch rlt = \i.\i^ (13). See Fig. 112. 149] SPECIAL CURVES. 389 The ovals of Cassini are defined byrr =:T^ Ex. 3. ? (14). h being a given constant. Since for a point P in SS thegreatest value of rr is <?, it follows that the curve will consist oftwo detached ovals surrounding S, S, respectively, or of a singleoval embracing both points, according as I; < Fig. 113. In the critical case of h = c the curve is known as the lemniscate of Bernoulli; this presents itself in various mathe-matical problems. If 0, the middle point of SS, be taken aspole, and OS as initial line, of a system of coordinates r^, 6^, we have r = r^ + c - 2cri cos 0^, r = r^ + c + ScTj cos 6i ; the equation of the lemniscate is therefore /.4 which reduces toCf. Art. 142. (,.j2 + cf - 4cVi« cos ^1 = C-,ri==2c=cos26li .. (15). Ex. 4. The magnetic curves. If S, S be the N. and S. poles of a magnet, the forces at anypoint P may be represented by /i//-^ along SP, and /n/r alongPS. A line of-force is a line drawn from point to point alwaysin the direction of the resultant force. Expressing that thetotal force at right angles to the line is zero, we have gsin,^ + ^,sin,^ = 0. 390 INFINITESIMAL CALCULUS. [CH. IX Idd 1 d6 „ ,, „, 7^V^=° (^^)- Hence, since r sin 0 = r sin 6, we have . ^d6 . .,d6 -sin S ^- + sin fl -r- = 0,o» as or cos 9 + cos & = const , (17). An eq
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