330 Mr. Youne on an Extension of a Theorem of Euler 
rows, as may also be inferred from the researches of Cauchy, hereafter men- 
tioned. 
It is proper to state, that the inferences marked (5), (6), (7), are analo- 
gous to, and were suggested by those of Sir William R. Hamilton, at pages 60 
and 68* of his “ Researches respecting Quaternions,” in the present Part of 
the Transactions; with a copy of which Researches I was favoured while the 
communication now before the reader (the above-mentioned articles excepted) 
was in the hands of the Academy. 
8. To these inferences we may add that three squares, multiplied by three 
squares, will produce three squares, provided a square in one factor have to a 
square in the other the same ratio that a second square in the former has to a 
second in the latter; for, in this case, as it is easy to see, an entire row of com- 
binations will disappear from the four-square form.f Similarly for seven 
squares, if three such equal ratios occur. And we may readily ascertain the cor- 
responding conditions for six and five. But the consequences of particular 
hypotheses of this kind will be more fully noticed in the following supplemen- 
tary observations. 
* Pages 258 and 266 of Vol. XXI., Part 2. 
+ The sum of three squares, multiplied by the sum of three squares, will also produce the 
sum of three squares, provided the factors be so related that the first row of results in the four- 
square construction vanish. This relation is furnished by the co-ordinates of the extremities of a 
system of semi-conjugates in an ellipsoid. * 
For, denoting these extremities by (2’, y’, 2’), (2” y!s 2"), (@””, y!”, 2”), it is a known pro- 
perty of the surface that 
(y” + afte + yl?) (2? + a? + zr) = (aly”’ — a2ly') + (a’'y!” — wivel)e + (aly! al ALP Foe 
and, consequently, from our four-square form, we must have 
; ‘ aly! + aityl! + ery! 
and similarly 
ala! 4 alle! 4. wl"2" = 0, 
and 
yz! + yz! + U0 = 
and thus is suggested a neat way of deriving several properties of the surface. But geometrical 
application is not our object in the present paper. 
[In the proof of this sheet I think it right to add, that these three equations are otherwise 
obtained by Mr. Weddle, in a paper on the Ellipsoid, published in the Cambridge Mathematical 
Journal for January, 1847, but not seen by me till my copy was in the hands of the printer. ] 
