183 



this adeuteric function to zero, we thereby oblige any one of the 

 ten vectors to terminate on a given surface of the second order, 

 if the other nine vectors be given. But it has been seen, that 

 the adeuteric vanishes, when any two of its ten vectors are 

 made equal to each other ; the surface which is thus the locus 

 of the extremity of the tenth vector, must, therefore, pass 

 through the nine points in which the nine other vectors respec- 

 tively terminate. On this account the ten vectors, or their 

 extremities, may be said to be, under this condition, homo- 

 deuteric, as belonging all to one common surface of the second 

 order. And thus we at once justify, by contrast, the fore- 

 going appellation of the adeuteric function, and also see that 

 to equate (as above) this adeuteric to zero, is to establish what 

 may be called the equation of homodeuterism, as in fact 

 it was so called in a recent communication to the Academy ; 

 while, as an abbreviation of the recent notation, we may now 

 write that equation as follows : 



S (±012345.6789) = 0; 



where the sum in the left hand member represents the adeu- 

 teric function. 



What has been shewn respecting the composition of this 

 adeuteric, may naturally produce a wish to possess some geo- 

 metrical rule for constructing the aconic function (012345), of 

 any six given vectors ; and the quaternion expression for that 

 function enables us easily to assign such a rule. For this pur- 

 pose, let a, b, c, d, e, f be the six points at which the six 

 vectors lately numbered as 0, 1, 2, 3, 4, 5 terminate, being sup- 

 posed to be all drawn from some assumed and common origin o ; 

 while G, h, i, k may denote the four other points, through which 

 the surface of the second order passes, when the equation of 

 homodeuterism is satisfied, and which are the terminations of 

 the four other vectors above numbered as 6, 7, 8, 9. The 

 aconic function, above denoted as 012345, of the six vectors, 

 oa, ob, oc, od, oe, of, which terminate generally at the six 



