admit of reduction in the number of Variables. 121 



form by which they admit of being expressed. For the proof of 

 the rule, only one additional remark has to be made in addition 

 to that already made, of the vanishing of the linear derivatives 

 necessarily implying the simultaneous evanescence of all the 

 other derivatives ; this additional remark being, that if the deri- 

 vatives of any class, linear or otherwise, qud one set of variables, 

 become all zero, the derivatives of the same class, qud any other 

 set of variables linear functions of the first set and the same in 

 number, will also become zero, for they are evidently expressible 

 as linear functions of the first set. 



Now let d^y d^, . . . d^-r be any {n—r) linear derivatives of U, 

 of which all the other of the v derivatives of this class are linear 

 functions, so that they vanish when these {n—r) vanish, and let 

 U be expressed as a function of {d^, d^, . . . dn-r) x^, Xc^. . . Xr)* 

 Then we may write 



U = </>^,0 + </>«i-l,l4-^m-2,2-f ••• +</>l,m-l-{-</>0, mj 



where in general <f>m-e,e denotes a function homogeneous and 

 of w — € dimensions in respect to d-^, dc^y . . . d^^r^ and homoge- 

 neous and of € dimensions in respect to x^, ^g, . . . x^. Now the 

 linear derivatives of U all vanish when c?j = 0, ^2=0 . . . df„_,.=0 

 for all values of x^, x^, . . . Xy- Hence U = on the same sup- 

 position, and hence </)o, ,„ is similarly zero. Also the first deri- 

 vatives of U, qud d^, d^, . . . dn-r, must vanish on the same sup- 

 position. Hence ^1,^-1 is identically zero; and so by taking 

 the 2nd, 3rd, ... up to the (m — l)th or linear derivatives of U 

 in respect to d^, dc^, . . . dn-ry we find successively 02, m-'i> <^3, m-zt 

 . . . (f>m-\, 3 each identically zero, and consequently 



U = ^rn, = </>(^l; d<^ ' ' ' dn-r)y 



as was to be proved. To express the fact of the y derivatives 

 being linear functions of {n—r) of them, form a rectangular 

 matrix with the coefficients of the v linear derivatives. This 

 matrix will be n terms in breadth and v terms in depth. Let 

 r=l: it is a direct consequence of the rule which has been 

 established, that every full determinant consisting of a square 

 (w) terms by [n) terms that can be formed out of this rectangular 

 matrix must be zero : again, let 7-=: 2; all the first minors, that 

 is to say, all the determinants composed of squares (?^— 1) terms 

 by {n—1) terms, must be zero, and so in general a loss of (r) 

 orders will require that the (r— l)th minors shall all vanish; if 

 rssri, the (n— l)th minors, i. e. the simple terms of the matrix 

 which are all coefficients of U, must vanish, or in other words, 

 when the function is of zero orders all the coefficients vanish (an 

 obvious truism). Thus, then, we see that the true rule for the 

 loss of one order in a polynomial of any degree is precisely the 

 same as the well-known rule for the loss of one order in a qua- 

 Phil. Mag. S. 4. Vol. 5. No. 30. Feb. 1853. K 



