Mr Maccullagh on the intensity of Light. 87 



latter ; so that in elliptic vibrations the intensity would be in- 

 dependent of the minor axis, which is far from being true. I 

 would propose the integral^/'?) - d t — so remarkable for its me- 

 chanical properties — as the measure of the intensity, the inte- 

 gral being extended to the whole time of a vibration. This 

 gives precision to the notion of vis viva, and leads moreover 

 to an elegant result ; for if a and b denote the semiaxes of the 

 ellipse, and T the time of vibration, the integral, by an easy 



2 cr - 

 calculation, will be found equal to — 7p- (d 2 + b-), showing that 



for the same colour the intensity is proportional to the sum of 

 the squares of the semiaxes, and that for different colours it in- 

 creases with the rapidity of the vibrations, as it would be natural 

 to suppose a priori. 



This theorem assigns very simply the reason why two por- 

 tions of light polarized at right angles do not interfere ; but 

 to show this, it will be necessary to lay down the following 

 general rule for compounding rectilinear vibrations having the 

 same period, whatever be the difference of their origins and 

 directions. 



Let AA' and BB', Plate I. Fig 3, bisecting each other at O, 

 represent in extent and direction the vibrations to be com- 

 pounded, and suppose C and D to be two simultaneous positions 

 of the moving molecule, which it would have in virtue of each 

 vibration singly. Complete the parallelograms O E and O J', 

 and through P describe an ellipse having O for its centre, and 

 touching the sides of the parallelogram O E ; this ellipse will 

 represent the resulting vibration ; it will have the same period 

 as the component ones, and equal areas will be described in 

 equal times about its centre. 



To apply this construction to the case proposed, it is ne- 

 cessary to show that when O A and O B are constant and at 

 right angles to each other, the intensity of the elliptic vibra- 

 tion, or the sum of the squares of the semiaxes, is independent 

 of the difference of origin, or of the position of the points C 

 and I). 



Now in an ellipse when a perpendicular from the centre on a 

 tangent makes an angle/ with the major axis, tin- square of its 

 length is equal to <j* cog* f I l> i" p. If f be the angle which 



