328 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Then, for a transformation of type (49), supposing the new K's to be 

 K,', K/, we have 



K,' = K, + IT S , 



K/ = K, + IT,, 

 and so 



K/ — K/ = K, — K s - (n, — n f ) < K ( - K s . 



Also, for a transformation of type (50), if the new El's are Uj IT/, we 

 have 



UJ = n. + K„ 



uj = U t + K„ 



and 



IV - uj = u s -u t - (k, - k.) < n 3 - n,. 



So when a condition of type (52) holds, any transformation applied will 

 reduce the difference of either the ITs or K's, if in fact it does not 

 change the sign of the difference. Further, the reduction is each time 

 by a value not infinitesimal, for it is at least 1 j st, as is seen by con- 

 sidering the values of K r and IT,, in (51). So the succession of trans- 

 formations of whatever kind must finally reduce the difference of either 

 the II's or the K's to zero, or change its sign, and then we secure either 

 condition a) or b). 



When one of these conditions has once been secured, any further 

 transformation will not change it; for, in condition a), a transformation 

 of type (49) will add at least as much to the K s as to the K„ and so 

 retain the inequality of the same order, and similar conditions are seen 

 to hold in the other cases. Also, as one of the conditions «) or b) must 

 hold finally, whatever the pair of values s and t, we shall have some 

 value as r such that 



n,. < IT S , K, < K„ s = 2, 3, m. 



from which follows the required condition 



Pr K Ps 



r ~~ z s 



9r < 9 1 



2, 3 m. 



11. The method of 6, applied to the surface resulting from the treat- 

 ment of 10, will secure the result of 6. It may be that already either 

 p r < r or q T < r, but in such a case the number of transformations of 



