278 Obituary Notices of Fellows deceased. 



This is called a memoir of pure geometry, and it is so in the sense that 

 no explicit analytical formulae are used in it. But Cremona clearly 

 saw that the propositions of Steiner which he was asked to prove 

 depended in great measure on properties of surfaces of any degree; 

 so he began with an outline of the general theory of surfaces, 

 assuming as known the fundamental properties of polar surfaces. Now 

 it is quite true that polar curves and surfaces can be defined in 

 geometrical terms, but this definition is artificial, and the question of 

 degree (which is essential in most of the applications) cannot be 

 decided, except by algebraical considerations, or by an extension of 

 Staudt's theory of imaginary points which is extremely laborious, and 

 not fruitful in results. Thus we cannot help feeling that the memoir 

 begins by a sort of evasion ; and every now and then we suspect the 

 author of having translated into a geometrical form a proof obtained by 

 analysis. Nevertheless, the memoir is a splendid contribution to 

 geometry in the proper sense; for even if analytical proofs were 

 supplied for the propositions most easily proved in that way, it would 

 not weaken the impression produced by the whole : we should still feel 

 that the writer is dealing with geometrical facts, and engaged in 

 geometrical speculation. The same may be said of Cremona's use of 

 the method of enumeration ; hazardous as it is in the hands of the 

 incompetent, he employed it with great effect in arriving at geometrical 

 conclusions : leaving to others the task of verifying his results in a more 

 rigorous way. In this he was following the example of Chasles, 

 Cayley, Salmon and others. 



Cremona did not again compete for the Steiner prize ; but it was 

 awarded to him in 1874 in recognition of his geometrical researches. 

 It was indeed well deserved ; for he had then published his treatises on 

 plane curves and on surfaces, as well as most of his papers of first-rate 

 importance. Among these are the researches, most closely associated 

 with his name, on the birational transformations of plane and solid 

 space, as well as of curves and surfaces. The most familiar example of 

 a birational transformation of a plane is that of inversion j this is a 

 particular case of the quadratic transformation which, in its normal 

 form, is 



x : y : z = y'z' : z'x' : x'y' 

 leading to 



x' : y' : z' = yz : zx : xy. 



The first consideration of this appears to be due to Magnus and 

 Steiner ; Cremona extended the method indefinitely by observing that 

 if we put x : y : z = < : x ' ^ where </>, x> ^ are polynomials in x', y', z' 

 of the nth degree such that, with a, b, c constant, a< + b\ + c^ = 

 goes through (n 2 - 1) fixed points, there is a one-one correspondence 

 between (x', y', z ; ) and (x, y, z). There is a similar theory for solid 



