223 



2 times, and so on, giving rise to the powers d lt 2 , ..... O a in the x 

 line. Similarly, form P # 2 , ... fo from the y line, and \^, // 2 . . . i// y 

 from the z line. Then the numerical part of the general term will 

 be 



n(L+/>)n(M+ g )n(N+r) 



D 



where in general Urn means 1 .2.3. ..m : as regards D, it is the fol- 

 lowing determinant, viz. 



L 3 M 3 N 3 



La M 2 N 2 



L! M, N, 



o+J9 V V 



v 

 X, 



The result, for greater brevity, has been set out in the above pages 

 for the case of S, a function of three variables, but the reader can 

 have no difficulty in extending the statement to any number. In 

 the case of a single variable, the formula can easily be identified 

 with that given by Burman's law. It is noticeable that the deter- 

 minant above written is of the form 



Apqr + Bpq + Cqr + Dqrp + Epe + Fq + Gr, 



the part independent of p, q, r being easily seen to vanish. More- 

 over, A, B, C, D, E, F, G, H are all essentially positive, so that D 

 can only vanish (except for p=0, r=0, y=0) by virtue of one con- 

 dition at least more than the number of the variables. 



VOL. VII. 



