468 



the surface from an umbilicar focus bears a constant ratio to 

 the rectangle under the perpendicuhir distances of the same 



brought it 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. Amyot'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 Examination 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 sui-faces 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 im- 

 portance of extending the signification of the terras focus and directrix, 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 ; 1 839, 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 enunciation, 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 prin- 

 ciple. 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 of the distances from the given planes, we 

 should not take the sum after multiplying the one square by any given positive 

 number, and the other square by another given positive number ; nor is there 

 any reason why we should not take other homogeneous functions of these dis- 

 tances. This conception would therefore be found of little use in geome- 

 trical applications ; while the modular principle, on the contrary, by employing 

 a simple ratio between two right lines, both of which have a natural connexion 



