8 KESEAKCHES ON H. 



of the cone, and AE its side; drawing from A a line AB perpendicular to the plane 

 of the circle, this will be the absolute altitude of the cone, or the distance of the 

 plane of the current from the pole, and from B drawing a perpendicular line BG 

 on the axis OY, we shall have BC = 2h 00= m. Also, drawing at ^a tangent 

 to the directrix, and from A the line AP perpendicular to it, this will be the alti- 

 tude of the elementary sector of the conic surface, whose base is ds; we must 

 determine the value of J. = AP. 

 The right-angled triangle AlBP gives 



AP' = J_B^ + BP\ 

 Let us draw from the centre of the circle the radius OE, which will be parallel to 

 BP and the two similar triangles OTE, BTP right angled at E and P give 



BP : OE : : BT : TO 



whence BP=^x BT=^{TO-BO) = OE-BO . ^; 



therefore ZP = W + (0E—^^^'\: 



OE 

 now, ^ = COS. TOE = COS. {FOE) = cos. (GOE—FOG) 



The angle GOE or its correspondent arc GEE is the distance of the element ds 

 reckoned from the origin of the arcs, on the circumference, representing that angle 

 by a, we shall determine the value of FOG in the following way. 

 "We have 



BO COS. (a — FOG) =B0 cos. a . cos. FOG + BO sin. a sin. FOG. 

 and from the right-angled triangle BOG 



OG = BO COS. FOG, BG = BO sin. FOG 



therefore BO cos. {a — FOG) = OG cos. a + BG sin. a 



= m cos. a + 2^ sin. u. 



Substituting these values in the preceding equations, and making the radius of the 



circle := B, we shall have 



A = AP=z Vr? + {R — m COS. u — j? sin. o)^ 

 Substituting also in the expression of r in (5) § 1, the values of the polar co-ordi- 

 nates of the circle 



(1) X ^ R sin. o, 



(2) y ^ R COS. (0, 



we shall have • 



(3) r' = R' + P — 2pR sin. o — 2mR cos. c, 

 where for the sake of brevity 



(4) P = m^ + n^ + f. 

 We have also in the case of the circle 



ds = Rda, 

 whereby the expression of the force (3) § 1, for the circular current will be 



, — li^R y^i^ + [R — in COS. (0 — p sin. u)^ . da 



[R' + l^ — 2 Rm COS. co — 2 Rp sin. cj)| 



