be taken from o tox, instead of from o to x. A being a function of r, x, and u, may be represented by 24 (r,x,u). To correct the fluent, let sm (fig. 23) = X, sm' x, then, the attraction of the solid, whose base is the quadrilateral figure prr'p', will be 24 (r, x, u) — 24 (r,X, u). In figure 24, call ps, u; sm, X; sm', x. The action of the solid, whose base is prr'p', is expressed by 24 (r,-x, u) · 24 (r,—X, u). In fig. 25, put psu, ps'=u', sm = s'm = x, tang. of rsmr: the attraction of the solid, whose base is the rhombus srs'p, on a point p in the produced diameter of the section, is 24 (1,x,u) — 24 (r,o,u) + 24 (r, — x, u') — 24 (r, — o,u′). Prop. C. Let fig. 26 represent the base or section of an infinitely long prism, and let this base be any right lined figure whatever, regular or irregular: from p, a point in the same plane, draw any line pq, cutting the base at s and m". It is required to find the action of the solid on the point p, in the direc tion pq. From the angles r, r', r", r'", &c. of the base, let fall the perpendiculars rm, r'm', r"m", r""m"", &c. on the line pq. Prolong the sides of the polygon till they meet pq at the points s, s', s", s'", &c. Put ups, u' = ps′, u" = ps", u"" = =ps"", &c.; and Also, let r = tang. rsm, x"= s′′m"} x"=s""m" X"= s′′m' r' = &c. =s"m" tang. r's'm', r" = tang. r"s"m", TM'"'= tang. r'"''s""m", &c. then it appears from the last proposition, that the attraction, of the upper half of the solid, is And in the same manner is found the attraction of the lower portion. If any part of the polygon, as pp', is parallel to pq, the attraction of that portion of the solid may be found by Prop. A. Scholium to Prop. 25, page 273. The following expression includes the attraction (on a point at the pole or vertex) of all this class of solids, where the generating plane is a regular polygon, and guiding curve a conic section: or where y2 = aa (ßx + yx2). Ваг A = 2n (x + arc (tang. = -—-√μx+vx3) — { (n − 2) 1+ya2 in which μ = ẞœa (1 + r2), = ßx* (1 + ra), v = 1 + yao ( 1 + ra), and (sine = , accordingly as v is positive or negative. Ex. 1. Let y=0, a = 1, y = ßx; in which case the solid is the polygonal parabolic conoid treated of in the proposition; and we have μ = ß (1 + r2), v = 1, whence μπ b2 x+nß }≈ +2nrßL the same as was found before. = 22, ß = a, y = — 1, y' — — ( ax — x') : a2 Ex. 2. Let a*= here the curve pr, fig. 15, is an ellipsis whose diameters are a and b, a being that which coincides with the axis pm. We the attraction of a polygonal spheroid, on a point at its pole, is b2 arc (tang. = 1⁄2-1⁄2 √21⁄2-2 (1+r) x + (1 — — (1+r))x°) (a2 — b2) √ a2— (1 + r2) b2 a2 arc (sine L accordingly as is greater or less than 1 + r2, or as — is b greater or less than the secant of half the angle formed at the centre of the generating polygon by one of its sides. When xa, the first arc in the above expression becomes simply arc (tang. =) (n—2) *, and we have for the action In like manner, may the action of the solid be found when the guiding curve is an hyperbola; the only difference between that case, and the one we have just considered, being in the value of y, which must be taken + 1 instead of — 1. Scholium to Cor. 3, Prop. 27, page 278. If the variable rectangle is given in species, and the touching curves are conic sections; that is, if 12 y* = x2 (Bx + yxa ), y12 = a12 (Bx + yx2), we shall have, for the action of the generated solid, on a point at its vertex by Prop. 4, A = 4√x arc (tang. = -√x+(1+1o) aa (ßx +yx2)) + 12 =√ + (1 + 112) ∞12 ( ßx + yx2)) — 2′′X, 4fx arc (tang. = a where r ==,r=; and by actually taking the fluent, a - 2πx, where μ = 2TX, ре arc (tang.=√x+vx3)· Bæ2 ( 1 + r2), v = 1+ yaa (1+r2), μ'=ßx2 (1+722), v' = 1+ ya'a (1 + p12), (sine 4r' Ba'z L + 1), or = arc ~ as v' is positive or negative. If, in the preceding expression, we maker and a′ infinite, andro, it is reduced to This is the action of an infinitely long cylinder on a point at the vertex of its transverse section, the equation of the said section being y2 = ao (Bx + yx2). Ex. If the base, or transverse section, is an ellipsis, or if b2 b2 y' = 22 (ax—x2), we have a2 = 3, ß=a, y = — 1; and a2 Scholium to Cor. 2, Prop. 30, page 281. If we would have a general expression for the attraction of such solids as the one we considered in the proposition, when the guiding curve is any conic section, or when y2 = x2 (ßx+yx'), there arises at first (from the formula for the action of a rhombus) √ x + ( 1 + ro ) a2 (ßx + √x2)) + A = 4√x arc (tang. =√x+ 4fx arc (tang. ÷ √x + (1 + ~13) aa (ßx + yx") = ́x* r2) — 27x, and by actually taking the fluent (tang. = —— √μ'x + v'x3) } − ( 430 42 + 2x) π + 0 + q', |