¢* ¢ 
ge t 4 Osa RnGy 1a 
?” 
¢ i 
*= 19" Tesan > + &e. 
1.2.3.4.5.6.7.8 
Again, 
ex =X, + Cy, + 0, 
where ,, 1, v, are the same functions of y that A, uw, v are of 
@. Hence we have 
eorex— A+ im +N, 
A standing for XA, + uy, + vy, M for Av, + wA, + vu, and N 
for Au, + pv, + vd, 
The three functions, A, Mm, Me N depending, each of them, 
on two variables, ¢ and yx, hold the same place in the present 
calculus that cosine and sine hold in the received trigonometry. 
And as the sum of the squares of cosine and sine is always 
equal to unity, so the equation 
A> + mM? + Nn? — 3AMN = 1 
holds good, no matter what be the amplitudes @ and y. 
The importance of these formule in our theory of triplets 
is most obvious. Fora triplet «+ vw + z may in general be 
thrown into the form m(A + ™ +N), which, as we have 
seen, is equivalent to me?+"x. So that if m be called the 
modulus, and ¢ and y the amplitudes, of the triplet, we shall 
find, on multiplying two triplets together, the following theo- 
rems to be true: 
The modulus of the product is equal to the product of the 
moduli of the factors. 
Either of the two amplitudes of the product is equal to the 
sum of the two corresponding amplitudes of the factors. 
The modulus m is connected with the constituents of the 
triplet (#, y, 2) by the following equation, 
mMa=a+ y+ 2— 3ayz, 
with respect to which it is to be observed that the right hand 
member is the product of 
