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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) = 05 
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 quaternton 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 0; 
_ while G, H, 1, 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, OF, OF, which terminate generally at the six 
