deduced from (1), (2) and (3), respectively, when the plus signs are taken, and (16), (18) and (20), those when the minus signs are used. The process by which (15) to (20) were obtained shows that the value of w is not altered by interchanging its coefficients in (7), (8) and (9). By inspecting these six equivalent formulas, we perceive that the square numerators will always be positive; and that they interchange by interchanging g, h, j, or O, P, Q, and changing the signs of the terms in the trinomials therein. If we change the signs of all the terms in the trinomial factors in the denominators of (15) to (20), in which Q, P, O, respectively occurs, in succession, each of these three general changes makes the trinomial factors, in each of the six denominators, identical in form, but the terms will be variously transposed. Hence, from any one of (15) to (20) all the others may be derived by permutation. Since all the signs of Qin (15) and (16), those of P in (17) and (18), and those of O in (19) and (20) are positive, it is obvious that when any one of these three quantities is greater than the sum of the other two, all the trinomial factors in the denominators of two of these formulas will be positive; but in the denominator of each of the other four formulas, two of said factors will be negative; then, because the product of two negative quantities is plus, all of the denominators, and thence w, also, will be positive. Therefore to make w positive, one of the quantities g, h, j, or of O, P, Q, must be greater than the sum of the other two. Take m = 2, in the first set of formulas for p, q, r, or m= = 4, n = 3, in the second set; then 5, p= 44, q=117, r = 240, a=125, b = 244, c=267, d= 161, e284 and ƒ = 357. = x == Substituting these in either of (15) to (20) we obtain w in terms of s, l, u, whose values are given; and then we know that = 44w, y=117w, z= 240w. These numbers will be very large, even when each of the quantities s, l, u, is equal to 1. Remarks:- Putting for a, b, c, d, e, f, any numbers, and canceling 8, l, u, (4), (5), (6), become [a2w2±dw=; b2w2±ew=☐; c2w2±ƒw=0;] . . . (21) then K=4a2b2cde, L= 4a2bc2df, M=4ab2c7ef, N=8a2b2c2, 0= a2b2f, Pa2c2e, Q = b2c2d. These values of K, L, M, &c. in either of (15) to (20), gives a formula so extensive in its application to the solution of those Diaphantine problems which require the fulfillment of six or more conditions, that we are surprised that neither Euler, Lagrange, Barlow, nor any other writers on the Diaphantine Algebra, so far as we know, has given it a place in his work. Substituting a for d, b for e and c for f, we get [a2w2±aw=; b2w2±bw=☐; c2w2± cw=☐.] .(22) This change makes K-4abc2, L = 4ab2c, M4a2bc, N = 8abe, O =ab, Pac, Q=bc. Taking the minus signs in (22) these values of K L M &c., in either of (15) to (20) furnishes a formula by which numbers may be found, at pleasure, that will fulfill the conditions in Dr. Perkins' first problem, on page 101 of the ANALYST; and using the plus signs in (22) we can find as many sets of numbers as we choose, which will satisfy the conditions in the Dr's second problem, on page 101, since in our formulas any numbers may be substituted for a, b, c, and those sought will be in Arithmetical, Geometrical or Harmonical progression, &c., if a, b, c, be so taken. If we put the first expression for v, in (C) and (D), in succession, in the first form of (10), those of (E) and (F) in that of (10′), and those of (G) and (H), in that of (10"), we shall have K=4gh, L=4gj, M=4hj, N=8, 0=j, P=h, Q=g. In these values of K, L, M, &c., g, h, j, may represent any numbers or their reciprocals. In all of the preceding cases they are put for the quotients obtained by dividing the coefficient of w by that of w2, in each of the given equations, and each of these quotients represents a fraction. Hence any numbers substituted for g, h, j, produce two values of w, and thence two sets of three square numbers. In all of the foregoing cases the signs must be taken disjunctively; but numbers ad libitum can be put for g, h, j, which will fulfill the six conditions in (7), (8) and (9), when the ± signs are taken conjunctively. In proof of this assertion, take for g, h, j, any three of the four numbers 2016, 3000, 3696, 4056, or any three of their reciprocals; then w= (65)❜ or its reciprocal, will satisfy all the conditions in (7), (8) and (9). We may also take for g, h, j, any three of the following thirteen numbers, or any three of their reciprocals; 103776, 231000, 369096, 566544, 582624, 867000, 936936, 1014000, 1068144, 1086336, 1172184, 1197000, 1215696; then w= (1105)2, or its reciprocal, will fulfill the said six conditions. No three numbers in (15) to (20) will produce either of these two values of w, they must be found by other methods of solution; therefore when g, h, j, are such as will fulfill the six conditions in (7), (8), (9) simultaneously, they will furnish three values of w, and thence six sets of three square numbers. [Soon after the publication of Prof. Perkins' solution of the "two similar Indeterminate problems," in the ANALYST, No. 6, Vol. I, we were requested to publish a solution of the same question, made some twenty years ago by Dr. David S. Hart, of Stonington, Conn. We wrote to Dr. Hart upon the subject and he sent us a copy of his solution, which is brief, and, we think, satisfactory. About some time we received the copy of Dr. H.'s solution, we received from Natreson, with a request to publish it, the foregoing solution of an ana, but somewhat more general problem. We had intended to publish both solutions in the same number, but, at present, our space wi. not permit. We hope to find room for Dr. Hart's solution in a future No.-Ed.] ་་ SOLUTION OF A PROBLEM. BY PROF. J Problem. - Pairs of angle; determine the loc WICKLIN, COLUMBIA, MISSOURI. ngents to any conic section intersect at a given of their intersection. Denote by (x, y) the co-ordinates of a point of intersection, and by (x'y') a point of contact. The general equation of any conic section, when referred to the principal diameter and tangent through its vertex, is y2 =mx + na2; therefore at the point (a', y') y'2=mx' + nx'2; (1) and for the tangent a of the angle formd by the axis of x, and a straight line through the points (x, y), and (x', y'), we have a = (y — y') ÷ (x — x'); whence • y' = y ax + ax'. Substituting this value of y' in (1) we obtain (a2 - n)x12 + (2ay · (2) (3) This equation gives two values for x'; but, since, by the conditions of the problem, the line through the points (x, y), (y', y') must have only one point, viz., (x'y'), in common with the curve, the two values of x', in (3), must be ‹ equal; . '. 4(a2 — n) (y2 — 2ary a2x2) = (2ay — 2a2x —m). Reducing this The two values of a in (4) belong to the two tangents drawn from the points (x, y). Denoting these two values by a', a", and the tangent of the angle which the two tangent lines make with each other by t, we get m2(y2 тх ・nx2) (4mx + 4nx2 + m2 + 4ny2)2 the equation of the locus required. If t = tan 90° = ∞, then 4mx + 4ny2 + m2 = 0 the equation of a circle. If n = 0, (6) (7) (8) the equation of an hyperbola. If t∞ in (8), 4x + m = 0, x = — ‡m, the equation of the directrix to the parabola. CRELLE'S JOURNAL. NOTE, by CHRISTINE LADD, Chelsea, Mass.-The last number of Crelle's Journal for pure and applied Mathematics opens with the continuation of an article by Stern on The Theory of Euler's Numbers. Sturm contributes a paper of forty pages on Cubic Curves in Space. Six points in space determine such a curve. Through five fixed points, quadratic curves can be passed, through four points, biquadratic curves. The author proposes to determine how many curves of each of these two classes will cut a given line, how many will pass through a given point, how many be tangent to the line, how many cut a plane, how many osculate the plane, and how many will satisfy at once all possible combinations of these conditions. Alsatia shows its mathematical activity by a paper of Milinowski's on A Reciprocal Relation of the Second Order. Given such a relation between three collinear systems; if a point or a line move according to a certain law in one system, according to what law will the corresponding point or line move in the other system? Mr. Mallet of Trinity College, Dublin, furnishes the next paper. Richolot has shown that, if R(x) represents a rational function of x and X= (x-x1) (x − x 2 ) . . . . (x — x2m), in which x1, x2 &c. are real quantities, positive or negative, the integral R(x)dx can be reduced, by a series of rational substitutions, to the normal form of hyperelliptic integrals. Mallet shows that the desired reduction can be effected by the union and repetition of only four substitutions. Th. Reye communicates an extension of a former article on Algebraic Surfaces apolar to each other. Cauchy and Abel have given a rule for the multiplication of two infinite converging series when the analytic moduli of the series are converging. F. Mertens, in the concluding paper of the number, shows that in order to the application of this rule, it is only necessary that the modulus of one of the series should be convergent. It is not greatly to the credit of the mathematicians of the vicinity that Crelle's Journal lies on the shelves of the Boston Public Library with uncut leaves; unless, indeed, we are to suppose that all who give themselves to mathematics are so rich that they can afford to take the Journal for themselves. Certainly none who hope to extend the boundaries of the science can afford to do without their Crelle. RECENT MATHEMATICAL PUBLICATIONS. COMMUNICATED BY G. W. HILL. Memoir on the Secular Variations of the Elements of the Orbits of the Eight Principal Planets. By John N. Stockwell, M. A. (Smithsonian Contributions to Knowledge. No. 232.) Washington, 1872. 4to. 200 pp. $2.00. An Investigation of the Orbit of Uranus, with General Tables of its Motion. By Simon Newcomb. (Smithsonian Contributions to Knowledge. No. 262.) Washington, 1872. 4to. 288 pp. On the General Integrals of Planetary Motion. By Simon Newcomb. (Smithsonian Contributions to Knowledge. No. 281.) Washington, 1874. 4to. 31 pp. |