GO 



1.2^1.2.3.4.5^ 1.2.3.4.5.6.7.8^ 

 Again, 



c'" = \ + iV/+ 'V, 



■where X^, /x„ v, are the same functions of ^ that X, fi, v are of 



^. Hence we have 



A Standing for XX^ + juju^ + w^, m for Xv, + n\ -\- vfij, and n 

 for XjU/ + fxv/ + vX;. 



The three functions, a, m, and n depending, each of them, 

 on two variables, ^ and t^^, 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^ + M^ + N^ — 3 AMN = I 



holds good, no matter what be the amplitudes <p and ;;^, 



The importance of these formulae in our theory of triplets 

 is most obvious. For a triplet a; + t?/ + i^z may in general be 

 thrown into the form ?w(a + tM + t^N), which, as we have 

 seen, is equivalent to jwe'* + '"''. So that if m be called the 

 modulus, and ^ and ^ 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 (x, y, z) by the following equation, 



m^ ■=. x^ + y^ + z^ — 3 xyz, 

 with respect to which it is to be observed that the right hand 

 member is the product of 



