262 Prof. Grassmann on the Theory of Compound Colours. 



represent the colour of the mixture, its direction showing the 

 tint, and its length tlie intensity of the colour. 



This done, the tint and intensity of any mixture of colours 

 may be found by mere construction. Thus it is only necessary 

 to determine the lines which represent the tint and intensity of 

 the mixed colours, and then to add these geometrically, that is, 

 to compound them as forces, and the geometrical sum (the 

 resultants of these forces) represents the tint and intensity of 

 the mixture. This follows immediately, because the order in 

 which the geometrical addition (or compounding of the forces) 

 is effected is without intluence upon the result. Thus the colours 

 represented in conformity with the above determination by the 

 lines a, b, a', b' may be taken as a basis ; then let ««, when « is 

 positive, represent a colour which has the tint a, and the inten- 

 sity of which is in the same relation to that of « as a to 1 ; or 

 if « be negative, a colour which possesses the tint of the com- 

 plementary colour a', and the intensity of which is in the same 

 relation to that of «' as « to 1 . The same applies to the second 

 colour b, and its complementary colour b'. Of the two colours 

 e and e', the component colours of which are to be ascertained, 

 let the first be represented by the mixture of the colours ua and 

 ^b, and the other by the mixture of the colours a'a and /3'Z», then 

 (leaving the intermixture of white out of the question) the mix- 

 ture of c and c' may be represented by the combination of the 

 four colours aa, ^b, a'a, /S'Z». But aa mixed with u'a gives 

 {u + a.')a, and yS6 mixed with /3'b gives {/3 + 0)b. Consequently 

 the mixture of c and c' may also be represented by the mixture 

 of the two colours {oi-j-u.')a and {^ + ^')b. As, however, these 

 latter have the original tints a, b or a', b', their mixture is re- 

 presented by the geometrical sum of the lines, and consequently 

 by the lines (« + «')«+ (/3 4-/3') 6, i. e. by (««-f-/3^») +(a'a + /S'Z>), 

 i. e. by the geometrical sum of two lines, which, taken separatelyj 

 represent the colours to be mixed. 



This law, which follows of necessity from the three original 

 assumptions, and which only requires a simple but complete 

 series of observations for the determination of the series of 

 colours, may also be expressed in another manner. Thus, if a 

 circle be drawn round the origin of the lines, having a for 

 its radius, and substituting for each line the point at which 

 it strikes the periphery provided with a weight proportional 

 to the length of the line, the mixed colour produced by two 

 given colours may be ascertained in the following manner. 

 Each of the colours is represented by a loaded point on the peri- 

 phery, in such a manner that the radius belonging to it shows 

 its tint, whilst the weight expresses the intensity, and deter- 

 mines the centre of gravity. A line drawn from the centre to 



