174 Prof. A, Anderson on some Properties of the 



It is interesting to note that expressions (9), (10), and (11) 

 reduce to (12) when a and p are each taken equal to unity. 

 In the limiting case of the Dual Comhustion Cycle, when 

 a and p each equal unity, the repeated cycle becomes one of 

 alternate compression and expansion along the same adiabatic 

 line. Clearly this case may be taken also as equivalent to the 

 Carnot or Constant Temperature Cycle between two infi- 

 nitesimally close adiabatic lines. This gives further evidence 

 of the approximate truth of the simple efficiency expressions 

 obtained in the paper. 



Some 'Properties of the Nul Point of Thin Axial 

 Pencils of Light directly refracted through a Symmetrical 

 Optical System. By Prof. A. Anderson*. 



IN a short paper in the Philosophical Magazine of January 

 last I showed that a point, which I now venture to call 

 the nul point, and which is easily found by experiment, could 

 be used with advantage in determining the constants of a 

 lens-combination. It may be defined as the point of inter- 

 section of the axis of a symmetrical optical system with those 

 axes perpendicular to it about which a small rotation of the 

 system has no effect on the position of the image. Rotations 

 in the same direction about perpendicular axes on opposite 

 sides of it produce displacements of the image in opposite 

 directions. It was proved in the paper just referred to that, 

 if be the nul point and P 1? P 2 the axial positions of the 

 object and image, OP 2 /OP! is the magnification. In other 

 words, all straight lines joining corresponding points of the 



Ffcr. 1. 



object and image pass through 0. It is, in fact, the centre of 

 perspective of the object and image, and changes in position 

 with a change in the position of the object. 



In fig. 1 the letters are those usually employed to denote 



* Communicated by the Author. 



