236 Sir Wrtr1am Rowan Hamitton’s Researches respecting Quaternions. 
Thus, in particular, the coefficient of the product m, mz in the expression thus 
obtained for mj’, is, 
Nooo = I" Log (coo Qo aif Con My) 
=F Lo, (Ca Ao af Con Gy). (197) 
The equations (188) permit us to write 
[og = An 3 ly, = — 3 — — Ay) 3 dy = Qo 5 (198) 
provided that we assign to / the value 
l= Ayy Ay, — Ao Ar (199) 
Hence 
— 4 (C000 Ao SF Com d,) — (Giro Any aia Con ay,) 9 
—— (200) 
Ayy Gy — Ao %M 
If we substitute, in this expression for 2, the values (186) for dy) G5 Qo Gp 
we shall thereby obtain, in general, a certain function of a,, a, which will be 
homogeneous of the dimension zero, because it will present itself under the form 
of a fraction, of which the numerator and the denominator will be homogeneous 
and quadratic functions of the same a, a, In order that this quotient of two 
quadratic functions of the number expressing the ratio of a, to a,, or of a, to a), 
may be itself independent of that ratio, we must have certain relations between 
the coefficients ¢,,,. &¢,, and the fraction itself must take a particular value con- 
nected with those coefficients; which relations and value may be determined by 
the three equations : 
Moo (Ero Cro — ©i00 Coo) = x9 (Coo + Coo Cow) 
— ,00 Goro (6:00 + Con) ; (201) 
Nooo (Coo Cin — S100 Con =5 Con Cro — F101 Coro) 
= n (ooo a Coon Cow) — 11 Coo (ooo =e Con) 
=F Cx Gon (E000 + Cond — 100 (oi Coo. te Cin) > (202) 
Nooo (Coo Cin — i Gay) = in Coo (CPs + Con) 
— €9) (Gio Con + Cu) (203) 
In like manner, each of the seven other coefficients, 7,,,, &c. in the expres- 
sions (196), will furnish three other equations of condition, which must all be 
satisfied, in order that the values of these coefficients of multiplication of couples 
