RxGKHS — Orf/ioffonrd Trajecfories of n Congriienr.c nf Curves. 113 



integrate on the assumption that «,, «2, n^, h, b^ are constant * the result will 

 be the family of curves defined by 



ttixAv + hychj + (u_ {ydx - xdy) = 0, 

 that is 



I log {ttiX" + b^y'') - ffj 



d'^ 



«, + bX 



X- 



These are two different types of curves according as aA which = is 



positive or negative. 



First, let a A. be positive and = t-. This may be named the e/H2)tic or sjnrnl 

 type. Since I = b, ~ ch. = - 2(h., it will be found that the curves form the 

 family 



- + - = /.-e*, 



pi ^2 



where 





These curves are evidently spircds round the origin ; they reduce to 

 equiangular spirals if pi = joj. Wlien 7=0, they are suddenly transformed 

 into ellipses — the Dupin indicatrices. 



SecondW, let or aj)-, be negative and = - t". This we call the 



hyperbolic or non-spiral type. The indicatrix will be the family of curves 



gi-p ,,1+p = constant, 



where ?, j, are the (real) factors of - + ^ and ;;=-—-=. Generally 



Pi P2 ^y-p,p2 



speaking, they resemble liyperbolas, s = 0, ij = being the asymptotes, but 

 they are unsymmetrical. 



Considering the indicatrix of either type as a family of curves lying in 

 the tangent plane at F, it has the following properties : — 



(a) If we proceed from F along any n-curve to a point Q, the tangent- 

 plane at Q, in the limit when Q approaches F, cuts the tangent plane at F iu 

 a line whose direction is that of the tangent line to any curve of the indicatrix 



* This assumption is justitifd by the fact that the integrated e(iuations give rise to the ditferential 

 equations, even if n,, fla, etc., me variable, the snuiU qaantilies of trie third ordei being neglected, 



