288 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
The negative term — B n + 1 ([m, 1] — l), it may be noticed, arises from decomposing 
the first term of [m -j- 1, n], as given by the original formulae, into two parts, of which 
it is one. 
Thus, ex. gr., 
+ \)(p + 2)(y + 3)(4p + 1) 
1.2.3.4.5 
is changed into 
(p - 1 )p(p + 1 )(p + 2 )(p + 3) p (p + 1) (p + 2)Q + 3) 
1 . 2 . 3 . 4.5 1 . 2 . 3.4 ^ ‘ 
The numbers in the hypothenuse of this infinite triangle, viz., 
1, 1, 2, 6, 36, 876, 408696, 83762796636, 3508125906207095591916, 
6153473687096578758445014683368786661634996,., 
are what I call the Hamiltonian Differences, or Hypothenusal Numbers* ; and their 
continued sums augmented by unity, viz., 
2, 3, 5, 11, 47, 923, 409619, 83763206255, 3508125906290858798171, 
6153473687096578758448522809275077520433167,., 
are what I call the Hamiltonian Numbers. The two latter of these have been 
calculated by means of Mr. Hammond’s formula, presently to be mentioned, and the 
corresponding Hypothenusal Numbers deduced from them by simple subtraction. 
Their connection with the theory of the T schirnhausen Transformation will be found 
fully explained in my memoir on the subject in vol. 100 of ‘ Crelle.’ My present 
object is to speak of the numbers as they stand, without reference to their origin or 
application, t 
* The other numbers of the “ triangle,” whose properties it may be some day desirable to investigate, 
may be termed co-hypothenusal numbers of order measured by their horizontal distance from the hypo¬ 
thenuse—their vertical distance below the top line denoting their ranlc. In the sequel the development 
is given of the half of a hypothenusal number (of the first order) in a descending series of powers (with 
fractional indices) of the half of its antecedent, the coefficients in the principal paid of such series being 
(not, as might have been the case, functions of the rank, but) absolute constants. These may be termed 
the hypothenusal constants. The values of the first four of them are shown to be 1, f, -p§-, 
t The reader will be disappointed who seeks in Hamilton’s Report any systematic deduction of the 
numbers which I have called after his name. He treats therein the more general question of finding 
the number of letters sufficient for satisfying’ any system of equations of given degrees by means of a 
certain prescribed uniform process whereby the necessity is obviated of solving any equation of a higher 
degree than the highest one of the given equations, and among, and mixed up with, other examples 
considers systems of equations of degrees 1, 2, 3; 1, 2, 3, 4; 1, 2, 3, 4, 5; 1, 2, 3, 4, 5, 6; for which the 
minimum numbers of letters required to make such process possible (when the equations are homo¬ 
geneous) are 5, 11, 47, 923, respectively. Accordingly he has no occasion to employ the infinitely 
developable Triangle which gives unity and cohesion to the problem which deals with an indefinite 
number of equations of all consecutive degrees from 1 upwards. This triangle, which plays an impor- 
