Important New TJieorem. 41 



sheet whose common intersection must not only contain all the 

 double points of the nomographic figures, but also all the inter- 

 changeable points; and which will .'. coincide when the figures 

 are in involution. 



Note. — The discussion of the equations for n odd and n even will be given 

 in a supplementary paper, containing an exposition of nomographic figures in 

 space, which may be termed " Uniquadric nomographics, " in as much as that 

 to points on any one of certain systems of quadrics, the correspondents are 

 on the same quadric* A geometrical exposition of such figures forms the 

 substance of an extensive Memoir of mine, read before the Royal Society of 

 London, in 1868, by the Rev. R. Townsend, F.R.S. : an abstract of which is 

 published in " The Proceedings " for that year. Sir William Hamilton's 

 solution of the problem is published in the "Transactions of the Royal Irish 

 Academy " for 1S49 and 1850. It is purely geometrical, but indirect; and 

 he has not given the " Quaternion " method which conducted him to his 

 results 



Aht. III. — Important New Theorem in the Geometry of Three 

 Dimensions, by Martin Gardiner, Esq., C. E., Member of the 

 Mathematical Society of London. 



[Read before the Society, June 2nd, 1869.] 

 If three quadrics have common intersection we may write their 

 equations as follows : — 



TJ + k" V = O; (1) 



TJ + k' V = 0; : (2) 



TJ + k Y = O; (3) 



Now, if o and o' be two fixed points, and m any point in the 

 quadric (3), then indicating by U, V ; U, V ; U, V ; the results 



m m oo o' o' 



of substituting the co-ordinates of m, o, and o in the qualities 

 U and Y, we have obviously the expression 



TJ + V Y 



U + k' V 



O 



as the ratio of the powers of the points m and o in respect to the 



