398 



Mr. E. J. Nanson on 



we get 



where f rs is given by 



A = | (a pq ) 



\ C rq) \Jrs) 

 — ) dpq | • | frs I ■> 



^q ^rq A-sq T «rs = Jrs" 



= 



Now, eliminating A, from ( 1 ) , (3) , we get 



whence f rs is given by 



I a pq I /m = 



(« w ) (fips) 



Vrq) "rs 



= e n 



by definition of e rs . Therefore, by (2) 



A = I a. 



I/; 



• (2) 

 . (3) 



which is the theorem to be proved. 



Sylvester speaks of this theorem as " one of the most 

 prolific in results of any with which I am acquainted," and 

 further says : " It is- obvious that without the aid of my 

 system of umbral or biliteral notation this important theorem 

 could not be made the subject of statement without an enor- 

 mous periphrasis, and could never have been made the object 

 of distinct contemplation or proof." 



The theorem just proved is a particular case of another 

 theorem given by Sylvester in the same paper. Before 

 giving a statement of this second theorem it is necessary to 

 explain a notation for a minor determinant and also for what 

 may be called a minor array. 



If 6, cf> be v-ads from 1, . . . n. then the minor formed from 

 the v rows 6 and the v columns cf> of the determinant 



I a Pi I (p i g=h--- n ) 



may be denoted by | a g( p | . 



In like manner, if 6 be a /tt-ad from 1, . . . in and <f> a v-ad 

 from 1, . . . n, then the array formed by the elements common 



