Prof. Cayley on Differential Equations and Umbilici. 441 



other oils, and of water, alcohol, ether, &c, were readily formed, 

 and rolled about upon its surface with great rapidity. 



In conclusion, I may perhaps be allowed to refer to a method 

 of producing the spheroidal condition of liquids in the cold 

 which was published by me, together with an illustrative wood- 

 cut, in a forgotten journal so long back as 1836. If a glass 

 goblet be about three parts filled with spirits of wine, and a 

 fiddle-bow be drawn across the edge, the usual crispating fans 

 from the vibrating segments will be produced ; but, in addition 

 to this, the spirit at the surface will be thrown into globules 

 which will be rolled upon the less agitated portions, and will 

 mark the nodes in the form of a four-, or six-, or eight-rayed 

 star very perfectly. These figures of Chladni may also be pro- 

 duced in globules with admirable sharpness by pouring turpen- 

 tine on water at about 150° in a goblet, so as to form a thin 

 layer. 



King's College, London, 

 November 12, 1863. 



LXIV. On Differential Equations and Umbilici {Sequel to the 

 Paper in the November Number, pp. 373-379). By A. Cayley, 

 Sadlerian Professor of Pure Mathematics at Cambridge*, 



IV. 



THE differential equation for the curves of curvature in the 

 neighbourhood of an umbilicus was obtained in a form 

 such as 



{bx+cy)(p*-l) + 2{f%+gy)p=0 ; 



and it was only because this equation did not appear to be 

 readily.integrable, that I considered, instead of it, the particular 

 form 



y(p' 2 — l)+2mxp = 0. 



But the general equation can be integrated ; and the result pre- 

 sents itself in a simple form. For, returning to the differential 

 equation 



(bx + cy){p*-l) +2(fa+w)P=<>, 

 and assuming 



bx + cy _ — 2v 



or 



fx+gy v 2 — 1 

 [bx + cy){v' l -\)^2{fx^gy)v = J 



* Communicated by the Author. 

 Phil. Mag. S. 4. Vol. 26. No. 177. Dec. 1863, 2 G 



