If the Prismoidal formula holds for this volume, we shall ascertain the fact by comparing the result of the integration of (1) with the result obtained by placing in the formula V = [Ao + 4A + Aa] (in which A is the section by the plane YZ, A that by the plane xa &c.) the values of the integral It is evident in the first place that the functions f(x) a..d (x, y) must be such that, on the integration of (1) with referenec to x, no new function shall be introduced different from what existed previously, such as a logarithm; for as this would not appear in (2), the results obtained by integration, and by the formula could not agree. It will be seen also that in (1) the result of the first two integrations must not contain radicals involving x, since on integrating again these would be raised to higher powers, which would not be done by the formula. Let us then assume 2= (x, y): (x, y) a rational integral expression of the form AxTMy” +BxTMy”-1 +Cx”-1y"+ .. + Exm+Fxm-1+ Gxm-2 + + Ky" + L′′-1 + Myn−2 + in which m and n are positive integers. (3) Since each term of this polynomial is to be integrated by itself we will consider only the term of highest degree Ax"y". Thus and as before let us consider only the first term A1; then xp(n+1)+mdx 0 Now let us apply the Prismoidal Formula. From equation (2) and similarly If however we consider only the term integrated in (6) we have ap(n+1)+m+1 + ..(8) 6 AA+1 The difference between the results obtained by the formula and by inte p(n+1)+m+1]. If the quantity in the parenthesis is zero the prismoidal formula must be correct. In the parenthesis let Thus it appears that the formula is perfectly correct so long as p(n + 1) +m does not exceed three, and that from this point on there is a constantly increasing error. Let us consider the cases that arise when different values are assigned .'. m = 0, = In the first Case f(x) = Ax3 + B122 + С12 + Q = y, and plane. 1 (x, y) = P = 2, thus the upper surface is a horizontal In the 2nd Case f(x) = A1x2 + B1x + Q = y, and 4(x,y) = A' x + P'z, a plane perpendicular to ZX. In the third Case f(x)= A,x + Q = y, and 4(x, y) = A'x2 + B'xy + Cy2 + D'x + E'y + P = z, an elliptic or hyperbolic paraboloid. In the fourth Case f(x) = Q =y, and (x, y) is a rational integral function of x and y in which y may appear with any exponent, but x must not be of higher degree than the 3rd. The surface represented by (x, y) may then be made, in this case, to pass through any number of points, provided that not more than four are in any single plane parallel to ZX. Let us now determine the limits of Weddle's Formula for seven equidistant cross-sections, a V = % [ Ao + 5A . + A2a + 6A3 + A4 + 5Aga+ A. ]. . (13) 20 6 6 a a 6 5AA3+1 @2(n+1)+m n+ 1 6p(n+1)+m AA^+1 2(n+1)+map(n+1)+ a 6 A2 = 6 n+1 6P(+1)+m &c.; hence ▲ An+1qp(n+1)+m+1 +2p(n+1)+m +6 × 3p(n+1)+m + 4p(n+1)+m + 5p(n+1)+m+1 + 6p(n+1)+m2 20X6P(+1)+m Subtract (6) from (14) and let the difference = ..(14) 8AA+1 ap(n+1)+m+1 n+1 2, 3. &c. Make p(n + 1) + m equal successively to 1, 86 = 113 699840 0, (= Thus the formula is exact when p(n + 1) + m does not exceed 5, and the error is small when p(n + 1) + m is 6, or even 7. The formula gives each term which is of too high degree a numerical value greater than that obtained by integration, but whether the final result is to be greater or less depends upon the signs of the coefficients of the terms of too high degree. Giving different values to p, we have, when p = 0, n any number whatever, and m = 5 at most if the result is to be exact. 1, we have f(x) = A1x + Q, (x, y) = Ax1 + Bx3y + Cx2y2 + Dxy3 + Ey1 + Fx3 + Gx2y + Hxy2 + Iy3 + Kx2 + Lxy + My2+Na+ Oy+P. If p = 2, If p = 3, f(x) = Ax2+ B1x+Q, and z= f(x, y) = Ax3 + Bx2 + Cxy + Dx + Ey + P. z=4(x, y) = Ax2 + Bx + P, a parabolic cylinder. If Ρ = 4 the upper limiting surface becomes a plane parallel to Y, and if p = 5; it is a plane parallel to XY. Let us now consider the wedgeshaped solid OABCDE, using x0p as coordinates instead of xyz, as shown in the fig is then 02 p=4(x,t) 1 α 2 0) v = S$S®; So=\2, dx.pd0.dp 'dx.pd0.dp = 1⁄2 S %S" ; [p2 ] ** [ ], da đôi. . . (15) 01 0 Let p = (x, 0) = v[xTMƒ{(0) + xTM-1ƒ1⁄2 (0) + ... ], ... considering as before only the first term we have 0 0. Now both the Prismoidal Formula and that of Weddle hold for this case to the same extent as for that treated before, since we have only to make p equal 0 in (6) to reduce it to a form similar to (16). Hence the prismoidal formula holds when x enters no term of (x, y) or 1(x, 0) to a degree higher than the third, and Weddle's when x enters no term to a degree higher than the fifth; while y or may enter in any form algebraic or transcendental, so long as the second integration is between constant limits. It is evident that these formulæ hold to the same extent for the areas of plane curves in ZX. Weddle's formula seems decidedly preferable to the Prismoidal for accuracy either for curves or solids. NOTE ON THE POLYCONIC PROJECTION. BY PROF. J. E. HILGARD, ASSISTANT IN CHARGE, U. S. COAST SURVEY. In an article which appeared in the last January number of the ANALYST (p. 15) Professor W. W. Johnson points out an error of statement respecting the description of the Polyconic Projection contained in the Coast Survey Report for 1853. In referring to this subject I merely wish to remark that it was thought a sufficient corrective to publish, together with a concise statement of the principles and formula of the polyconic projection, and extended tables, a map showing the whole globe on this projection on which the deviations from a right angle of the intersections of meridians and parallels are brought directly before the eye. This was done in the Coast Survey Report for 1856, plate No. 65, a reduced copy of which, herewith sent, may be interesting to your readers. In the subsequent Report for 1859, the diagram of the whole sphere was again given with a representation of the isogonic lines. These illustrations appear to have escaped the notice of Prof. Johnson. |