On the Surfaces of the Second Order. 281 



distance of any point of the surface from an umbilicar focus 

 bears a constant ratio to the rectangle under the perpendicular 

 distances of the same point from two directive planes drawn 

 through the directrix corresponding to that focus; and it is 

 easy to see that this ratio, the square root of which we shall 

 denote by //, is equal to L - M, or, neglecting signs, to the. 

 sum of the numerical values of L and M. Of course, if the 

 distances from the directive planes, instead of being perpendi- 

 cular, be measured parallel to any fixed right line, the ratio 

 will still be constant, though different. For example, if the 

 fixed right line for each plane be that which joins the corre- 

 sponding umbilic with either focus of the section xy, the ratio 



forward. Mr. Salmon had in fact proposed it for investigation to the students of 

 the University of Dublin, at the ordinary Examinations in October, 1842 ; and it 

 was published, towards the end of that year, in the University Calendar for 1843, 

 some months before the date of M. Cauchy's report, by which the contents of 

 M. Amy of s memoir were first made known. The parallelism of the two given 

 planes to the circular sections of the surface is also stated in the Calendar ; but this 

 remarkable relation is not noticed by M. Amyot, nor by M. Cauchy (see the " Exa- 

 mination Papers" of the year 1842, p. xlv., quest. 17, 18 ; in the Calendar for 1843). 

 It is scarcely necessary to add, that the analogue which M. Amyot and other 

 mathematicians have been seeking for, and which was long felt to be wanting in 

 the theory of surfaces of the second order, is no other than the modular property 

 of these surfaces, which appears to be not yet known abroad. M. Poncelet insists 

 much on the importance of extending the signification of the terms focus and direc- 

 trix, so as to make them applicable to surfaces ; and he supposes this to have been 

 effected, for the first time, by M. Amyot. These terms, however, applied in their 

 true general sense to surfaces, had been in use, several years before, among the 

 mathematical students of Dublin, as may be seen by referring to the Calendar 

 ("Examination Papers" of the year 1838, p. c. 1839, p. xxxi.). 



The locus above mentioned, being co-extensive with the umbilicar property, 

 does not represent any surface which can be generated by the right line, except 

 the cone. To remedy this want of generality, M. Cauchy proposes to consider a 

 surface of the second order as described by a point, the square of whose distance 

 from a given point bears a constant ratio either to the rectangle under its distances 

 from two given planes, or to the sum of the squares of these distances. This enun- 

 ciation, no doubt, takes in both kinds of focals, and all the species of surfaces ; but 

 the additional conception is not of the kind required by the analogy in question, 

 nor has it any of the characters of an elementary principle. For the given planes, 

 according to M. Cauchy's idea, do not stand in any simple or natural relation to 

 the surface ; and besides there is no reason why, instead of the sum of the squares 



