166 Mr. A. Cayley on Curvature [Feb. 13, 



oration. 



Declination. 



JJip. 



Horizontal lorce. 



Intensity. 





, , 17-959 



66 3 -975 



3-9613 



10-1271 





17-756 



66-678 



4-0028 



10-1103 



Ghent 



17-823 



67-221 



3-9197 



10-1232 



o ... 



16-233 



66-464 



4-0145 



10-0522 







66-948 







Louvain 



16-824 



66-898 



3-9565 



10-0828 



Mechlin 





66-714 









17-216 



66-573 



4-0065 



10-0767 





17-541 



66-538 



3-9941 



10-0311 





18-097 



67-211 



3-9152 



10-1077 



Spa 



16-627 



6Q-653 



4-0239 



10-1531 



Tournay 



17-691 



66-632 - 



3-9975 



10-0776 



Tronchiennes .... 



17-867 



67-361 



3-9032 



10-1397 



Turnhout 



17-025 



67-113 



3-9542 



10-1665 







66-718 







Secular variation . 



-0-1255 



-0-0573 



+ 0-00542 



-0-01155 



February 13, 1873. 

 Rear-Admiral RICHARDS, C.B., Vice-President, in the Chair. 



The following communications were read : — 



I. " On Curvature and Orthogonal Surfaces." By A. Cayley, 

 F.R.S., Sadlerian Professor of Mathematics in the Univer- 

 sity of Cambridge. Received December 27, 1872. 



(Abstract.) 



The principal object of the present Memoir is the establishment of the 

 partial differential equation of the third order satisfied by the parameter 

 of a family of surfaces belonging to a triple orthogonal system. It was 

 first remarked by Bouquet that a given family of surfaces does not in 

 general belong to an orthogonal system, but that (in order to its doing 

 so) a condition must be satisfied : it was afterwards shown by Serret that 

 the condition is that the parameter considered as a function of the coor- 

 dinates must satisfy a partial differential equation of the third order ; 

 this equation was not obtained by him or the other French geometers en- 

 gaged on the subject, although methods of obtaining it, essentially equi- 

 valent but differing in form, were given by Darboux and Levy; the 

 last-named writer even found a particular form of the equation, viz. what 

 the general equation becomes on writing therein X=0, Y=0 (X, T, Z 

 the first derived functions, or quantities proportional to the cosine-inch- 



