438 THE THE011Y OF SCEEWS. [400 



It follows that D -T- A and (B + C) -=- A must both be independent of 8. We 

 may therefore make 



and thus the homographic equation becomes 



where A is the only quantity which involves 8. 



The equation can receive a much simpler form by taking the infinite 

 objects as the two originating objects from which the range was determined. 

 In this case the equation 



must have as roots X = and X = x, and therefore 



4=0; D = 0, 



hence the homographic equation reduces to 



B\ + C/* = ; 



since B -=- C is a function of the intervene 8, we may say, conversely, that 



We have, however, already learned that the intervene is to have the 

 form 



Now we find that it can also be expressed, with perfect generality, in the 

 form 



It follows that these two expressions must be equal, so that 



In this equation the particular value of B does not appear, nor is it even 

 implied. The formula must therefore represent an identical result true for 

 all values of X and all values of /it. 



We may, therefore, differentiate the formula with respect both to \ and 

 to fji, and thus we obtain 



