58 



5. Its vector angle, p, is the angle between the radius 

 and the symmetric axis, 



6. The amplitude, lo, of the triplet (x, y, z) is the angle 

 between the modular plane and the plane containing the radius 

 and the symmetric axis. 



We are able to determine the radius, vector angle, and 

 amplitude, of the triplet (r.2, p-2, oJi)) which is formed by mul- 

 tiplying together two triplets (r, p, w) and (?-i, pi, wi), by means 

 of the following equations, in which s stands for the angle be- 

 tween the symmetric axis and the positive portion of any one 

 of the axes of coordinates. 



rri cosp coSjOi= r2C0ss coSjOj (1) 



?•/•, sinp sinp, — r^ sins sin p2 (2) 



b) + (1)1 = W2' (3) 



It appears from the first two of these equations that whe- 

 ther we call ■ -, or — ; — -, the modulus of the triplet, it 



coss sins 



will be true to say that the modulus of the product is equal to 

 the product of the moduli of the factors. 



The third equation asserts, that the amplitude of the pro- 

 duct is equal to the sum of the amplitudes of the factors. 



hs, the real unit is supposed to be placed on the axis of x, 

 we shall obtain the following theorem by dividing the second 

 of the preceding equations by the first : 



The tangents of the vector angles of the real unit, of the 

 two factors, and of the product, form a proportion. 



Mr. Graves stated that Sir Wm. Hamilton had been the 

 first to announce that if the real unit line, the factors, and the 

 product line, be projected upon the symmetric axis, the pro- 

 jections will form a proportion in the simple sense of that 

 term : whilst the projections of the same lines on the symme- 

 tric plane form a proportion, according to the higher sense in 

 which Mr. Warren uses the same word. 



The former of these theorems is merely the geometric in- 



