PROBLEMS OF MODERN SCIENCE 



polar, or otherwise, and even when we have decided 

 to use, say, Cartesian co-ordinates, we can choose 

 them in any two perpendicular directions what- 

 ever. The expression which represents the total 

 energy can thus take a variety of forms, but it is 

 always numerically the same. 



Now all the ordinary conceptions of geometry, 

 for instance, have some intrinsic character (or 

 more than one) which is often expressed by the 

 fact that something involved in the conception has 

 the same value whatever axes of co-ordinates are 

 chosen. For instance, an ordinary ellipse, referred 

 to any two perpendicular axes through its centre, 

 has an equation of the form 



ax 2 + 2hxy + by* = i 



i.e., any point on it, of co-ordinates x and y, 

 satisfies this equation. If we choose another pair 

 of axes, at any angle to the first, the ellipse has 

 a new equation, say 



A* 2 -f 2Hxy + B/=i 



where the large letters exhibit no obvious relation 

 to the small ones. But we can prove that, always, 



A + B = a + b 

 AB - H 2 = ab - K 



So that (a + b] and (ab h?) are always the same 

 for the same ellipse, and we call them the invariants 

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