The Messenger of mathematics Volume 1 - Tapa blanda

Reseña del editor

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. 1872 Excerpt: ... by taking c = c'. Trace the two curves y=-(e + e s),-x i---shewn respectively by the black line and the dotted line in fig. 9. Draw any line parallel to the axis of x, meeting the first curve in the points P, P' respectively, and let the ordinates MP, MP' meet the second curve in the points Q, Q respectively; then it is clear, that if for a given value of I the line PP can be drawn in such wise that MQ + M'Q' = I, there will be in fact the required two values C= OM and C'=OM' And since for MP very large we have MQ, and therefore also MQ + M' Q very large, and as MP diminishes, MQ + M'Q' also diminishes until it attains a certain minimum value, say =, it is clear that if I has any value greater than this minimum value, PP' can be so drawn that QM+ Q'M' = I. The above remarkably elegant investigation in regard to the two values C, C was given in the Examination; it seems to be the case that as PP' moves downwards, MQ+M' Q' continually decreases (viz. MQ decreases more rapidly than M'Q' increases), its value being least, and =2NS when PP' becomes a tangent to the first curve at its lowest point R; but it is not by any means easy to prove that this is so. The question depends on the form of the curve defined by the equations X = xl £-e-Z) + Jcc, (4-e£t), a a a a Y= xl (e + e-2) = fa (e? + e-s), where Xand Fare the current coordinates. and the equation may be written d. v'.,-dA „, dr „, dp eft eft r eft rK (ft r whence v3--in-j-= C-2 lffrdr-2 JTfrpdp, the required result. 12. Ifx,y are the coordinates of a particle moving in piano under the action of a central force varying as (distance)': write down the expressions of the coordinates «, y in terms of the time t and of four arbitrary constants: and (in case of disturbed motion) starti...