Proceedings of the London Mathematical Society Volume 22 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1891 Excerpt: ...X terms containing to be investigated is COG Hence, dividing numerator and denominator by (c--y)' and then putting c = y, the fraction becomes 1/36J2, and this does not vanish. daf The other factor of the third term, has been shown above oxccj not to vanish necessarily. Taking for /the form B (c--yOCc--ya)(c--ys), it follows that 5-5-= 5-5-0-yi)(c-ys)0--yj) da-dc ctedc-(-vOC--Cc-rOC-rO-l'Cc-yc-y, ) + ic-71)(c-ys) + (c-y, )(c-yi) + (c-y, )(c-y2) +ie-i(2C-yj-y, )-(2C-y, -yi)-(26-ri-yi). Hence it would appear that, when c = yi = 7j = y-i-- -would vanish also; but this is not necessarily the case, for ylt y2, y3 are irrational functions of a; and y. Hence their differential coefficients may, and in some cases do, become infinite at points on the node-locus, does not necessarily vanish on the node-locus. In like manner the term j may not vanish on the j, da; caidc, node-locus. 1 Sc Similar remarks apply to the terms containing-y2. (This point is specially considered in Example 4, below.) Hence does not vanish at points on the node-locus; but IB, ------do vanish at CX Co? points on the node-locus. Hence, if N = 0 he the equation of the node-locus, N occurs as a factor in S. Hence it must be possible to satisfy at the same time (18), (19) and As this is identically satisfied, it follows that the equations (18), (19), (20) can be satisfied by common values of x, y. The common values of y are given by equation (28). They cannot be such as to make the differential coefficients with regard to x, y of 'iax+gy = 0 vanish, for this curve has no singular points; and the only singular point on the curve (18) is 2a a Now, the common value of y, given by equation (28), is Cjl_ 4-bg+9a, 2(2&-9a) #2(2&-9a), hg1 a-46ff+46-30a, 8(6a-6) f/8(6a-&), 2bg + Sa If t