586 Mr. J. J. Sylvester's Note on Barman's Law for 



man's general formula. I recently have had occasion (as a prelimi- 

 nary step to the investigation of the laws of inverse transformation 

 between two systems of t variables each, instead of between two 

 single variables only, an investigation in which I have already 

 made such progress that I expect shortly to be in possession of 

 the general formula for the purpose) to reconsider what I shall 

 term Burman's law, and have been somewhat surprised to find, 

 that so far from affording a complicated expression, it does, when 

 properly stated, give rise to an expression of the very simplest 

 form that could be conceived or desired, and one that admits of 

 an easy and elementary proof. 



(Pu 

 To fix the ideas, let us take the case of -j-y, where x=<j>u. 



d^x 



dx^ 



For greater brevity write — . as Xr. The most cursory consi- 

 deration will suffice to show, irrespective of all calculation, that 

 we should have the following form of expansion, viz. 



+ { (2, 6)^2 . ^6+ (3, 5) . (^3 . ^s) + (4, 4) . (0,-4 . ^J } -^x,^ 



~ {(2,2,5)a72.a72.a?5+ (2,3,4) .(a?2.^3-^4) + (3,3,3)fe.a73.iP3)-^a?iio 



+ {(2, 2, 2, 4)(a72 . ^2 . a^a . x^) + (2, 2, 3, 3){x^ . x^.x^.x^)} -r-a?/* 



— { (2, 2, 2, 2, 3) {x^ . a^g . ^2 • ^2 • ^5) "^^1^^} 

 + (2,2,2,2,2,2)-^^^ 



In the first group of a single term, 7 is taken in one part, in 

 the second group of 3 terms, 8 is taken in every possible way of 

 partition in two parts, in the third group of 3 terms, 9 is taken 

 in every possible way of partition in three parts, and so on, until 

 finally 12, i. e. the double of the number next inferior to the 

 given index 7, is taken in the sole possible way in which it can 

 be taken of six parts ; I ought to add, that in the groups of 

 indices, unity is always understood to be inadmissible. 



The groups of indices in the parentheses indicate numerical 

 coefficients to be determined, and the whole and sole real diffi- 

 culty (if any) of the question consists in determining the value 

 of these numerical symbols. Now the law which furnishes these 

 values would be seen on the most perfunctory examination to be 

 the very simplest law that could possibly be stated, viz. any such 

 symbol as (r, s, t, . . .) is to be understood to denote the number 

 of distinct ways in which a number of things equal to the sum 

 of the indices r, s, t, &c. admit of being thrown into combina?^ 

 tion groups of r, s, t, &c. I 



