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theorems relating to the principal circular sections of a cen- 

 tral surface of the second order, and the sphero-conics traced 

 upon it. 



In a central surface of the second order, the arc of a prin- 

 cipal section P, included between the two principal circular 

 sections C, C, is bisected at the point t, where it touches a 

 sphero-conic of the surface. 



The proof of this theorem is extremely simple. The serai- 

 diameter drawn to the point t is obviously a semiaxis of the 

 section P; and the semidiameters drawn to the points cand c, 

 in which P meets C and C, are equal to one another, being 

 each equal to the mean semiaxis of the surface ; consequently 

 the arc cc is bisected at t. Precisely in the same manner it 

 might be shown that 



The arc of a diametral section, included within one of 

 the sphero-conics of the surface, is bisected at the point where 

 it touches a second sphero-conic of the same system. 



From the theorem first stated we deduce the following : 

 The sector of the surface included between the two principal 

 planes of circular section, and any diametral plane which 

 touches a fixed sphero-conic, is of constant volume. 



For, if we draw a second diametral plane P', infinitely 

 near to P, and touching the same sphero-conic, the two ele- 

 mentary sectors respectively included between P, P', and each 

 of the two principal planes of circular section, will evidently be 

 equal : and for this same reason 



The sector included between a cone generated by a semi- 

 diameter moving along one of the sphero-co7iics of the surface, 

 and any diametral plane which touches a fixed sphero-conic, is 

 of constant volume. 



