If in these equations we write Bi for B, X continues a real function, viz. we have X = (1 − x) (1 + x tan2 a) (1 + x tan2 ßi) (1— x tan3 a tan3 Bi) ; where observe that B+ C, = √1 − sin2 (a + Bi) sin2 + √1 − sin2 (a — ẞi) sin3 & = is real; viz. these formulæ give the values of аф √√1-sin2 (a + Bi) sin2 + in terms of the integrals аф √1 — sin2 (a — ẞi) sin❜ I day and √x dx dx √xX Nad We may change the form by writing tan ẞi = i sin y, whence X = (1 − x) (1 + x tan2 a) (1 − x sin3 y) (1 + x tan2 a sin2 y), sin2 = and observing the equation x cos2 y = (1+kx) (1 + Xx) ̄ ̄ ̄ (cos2 a +x sin2 a) 1 − x sin2 y we see that to real values of there correspond values of x which are positive and less than 1, and that as a passes from 0 to 1, sin passes from 0 to 1, or o from 0 to 90°, X being thus always real and positive. Writing siny, the relation between p, x gives a relation between x, y: viz. this is viz. this is a quartic curve; and introducing z for homogeneity, or writing the equation in the form we see that y3 (z + xx) (z +λx) − (1 + x) (1 +λ) xz3 = 0, x=0, z=0 is a fleflecnode, the tangents being z+x= 0, 2 + λ = 0; y=0, z=0 is a cusp, the tangent being y = 0; x = 0, y=0 is an ordinary point, the tangent being x = 0; hence the curve, as having a node and cusp, is bicursal. 467. The transformation of a given imaginary modulus into the form sin (z+ẞi) presents of course no difficulty: assuming that we have k=e+fi, then we have to find ɑ, ß such that e+fi=sin(a+ẞi), or writing sin a = §, sin ßi = in, to find, n from the equations C2 whence e2 +ƒ2 = §2+n2, and thence easily = If as above sin Bi-i tany, then tan y=n, or the equations give §, = sin a, and ŋ, = tan y. dx 465. The integrals fan, fade NP' NP are also reducible to elliptic integrals when the quintic function P has the form P = x (1 − x) (1 + xx) (1 + λx) (1 + x + λ + xλ x), as shown by Prof. M. Roberts in his "Tract on the Addition of Elliptic and Hyper-elliptic Integrals," Dublin, 1871, p. 63; and in the Note, p. 82, to the same work, a simple demonstration is given of the theorem (due to Prof. Gordan) that the like integrals, wherein P denotes a sextic function the skew invariant of which vanishes, are reducible to elliptic integrals. ADDITION. FURTHER THEORY OF THE LINEAR AND QUADRIC TRANSFORMATIONS. The Linear Transformation. Art. Nos. 469 to 473. 469. WE consider the transformation of the differential expression dx √x-a.x-ẞ.x-y. x-d' where the new variable y is given by an equation of the form xy + Bx + Cy+D = 0. The coefficients B, C, D might be expressed in terms of any three pairs of corresponding values of the variables x, y, say the values a, ẞ, y of x, and the corresponding values a′, B, y' of y: but it is better to consider in a symmetrical manner four pairs of corresponding values, viz. the values a, B, y, 8 of x and the corresponding values a', B', y, & of y. We have thus four equations from which B, C, D may be eliminated, and we obtain the relation which in fact expresses that the two sets of values (a, B, y, 8) and (a, B, y', ') correspond homographically to each other. 470. Writing for convenience a, b, c, f, g, h = B−y, y-a, a - B, a-S, B-d, y − d, a', b', c', f', g', h' = ß' — y', y' — á', a' – B', a' — S', B' − d', y'− d', so that identically af+bg+ch =0, a'ƒ' + b'g' + c'h′ = 0 ; then, as is well known, the relation in question may be expressed in the several forms af bg cha'f' : b'g' : c'h' ; or, what is the same thing, there exists a quantity N such 471. The relation between (x, y) may now be expressed in the several forms, and writing for (x, y) their corresponding values, the values of P, Q, R are found to be ƒ3PN2=ƒ'2QR, g2 QN2=g"RP, h2RN2=h"2PQ,√PQR f'g'h' 472. Differentiating any one of the equations in (x, y), for instance the first of them, we find a.x-B.x-Y (x−8) √x-8 √y — a' . y — B' . y — ' _ √ P Q R √ x − a . x − B . x − y (y-8') Ny-8' Ny-ay-B'.y-y-√PQR Jx - 0 -α.x-В.x-y.x-d or if we please |