186 



trary case. And the aconic and pyramidal functions having 

 thus been separately constructed they have only to be com- 

 bined with each other, according to the law already stated, in 

 order to assign a geometrical signification to each term of the 

 adeuteric function, namely, the sum, 



2 (± ABCDEF . GHIK) J 



and also to the equation of homodeuterism, which may now be 

 written thus (as in a recent communication to the Academy), 



S (+ ABCDEF . GHIK.) = 0, 



and which expresses that the ten points, a, b, . . . k, are situ- 

 ated upon one common surface of the second order. And if we 

 place the arbitrary origin o at one of the ten points, the num- 

 ber of terms in the adeuteric function, or in the equation of 

 homodeuterism, is easily seen to reduce itself, then, from 210 

 to 84. 



If the thirty co-ordinates of the ten points were substituted 

 in the function above called the adeuteric, the resulting expres- 

 sion could doubtless only differ by some numerical coefficient 

 from that determinant which might otherwise be found, as the 

 result of the elimination of the nine coefficients A, B, C, D, 

 E, F, G, H, I, between the equations, 



Ax 2 +By 2 +Cz 2 +Dy z +Ez x +Fx y + Gx Q +Hy +Iz Q + 1=0, 



Ax 2 9 +By 2 9 +Cz% + Dy 9 z 9 +Ez 9 x 9 +Fx 9 y 9 + Gx 9 +Hy 9 +Iz 9 + 1 =0. 



And Sir W. Rowan Hamilton has much pleasure in re- 

 ferring to a paper by Mr. Cayley, printed near the commence- 

 ment of the Fourth Volume of the Cambridge Mathematical 

 Journal, on Pascal's Theorem considered in connexion with de- 

 terminants, which paper had not been noticed by the present 

 writer, till his attention was called to it by a friend to whom' 

 he had communicated the above-stated construction. But 



