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XIX. — On Linear Constructions^ hy Rev. Thos. Penyng- 

 TON KlEKMAN, A.M., RectOT of Crofi-with-Southworth. 



Read March 18, 1851.. 



It is generally, known to mathematicians, and is stated by 

 Professor Chasles in his " Apergu Historique, &c.," as well 

 as by Professor Steiner in his " Systematische Entwickelungy 

 u. s. ?o." (Anhang), that the following question was twice 

 proposed as a prize question by the Academy of Brussels 

 above twenty years ago, and received no answer : " What 

 is the relation among ten points of a surface of the second 

 degree?" One obvious answer is, that the equation to the 

 surface, if the constants are made functions of nine given 

 points, expresses the required relation among the co-ordi- 

 nates of ten points. But it is plain that the solution re- 

 qviired is to be purely geometrical, and such that it shall 

 give a criterion independent of all properly analytical re- 

 sults, computation of numbers, or measurement of distances, 

 whereby it may be determined whether any tenth point 

 lies on the surface which passes through a giveu nine, and 

 where any line meets the surface a second time. 



Mr. Weddle, of the Royal Military College, Sandhurst, 

 gave a construction of the tenth point in the November 

 number 1850 of the Cambridge and Dublin Mathematical 

 Journal; and his is the first answer, so far as I can learn, 

 that the Brussels prize question has received. I am not sure, 

 nor does Mr. Weddle seem to be certain, that his solution 

 ■will be accepted as purely geometrical, for it is the con- 

 struction of an analytical result. Further, I am not able to 

 say whether the Brussels Academy will grant the use of the 

 compasses in the required construction, and Mr. Weddle 

 does not show that his can be effected without them. 



