Mr. R. Hargreaves on the Tgnoraiion Problem. 487 



viewed as a system in which oo v and % m+r are treated as 

 known quantities, while % p and i ra + r are to be found. The 

 value of Mm+r is given by 



s p 



and therefore 



r r r, s 



— ^ <^&m+r -^ @>m-\-s,p -"-m+r,m+s* 



The interpretation of the summation %a m+S}P A m+r , m+s or 



s 



2 <%,»*+« A m+r>w+5 , is that in the determinant A is to be sub- 

 stituted a row beginning with a p , m+ i in lieu of the row 

 beginning with a m+rj7n+ i, viz. 



Vp, 7n+r' 



nn + l, m+l 



p,m + l 



U/i . a 



The value of % p is given by 



Af p =2 b Piq w q +'2 l c Pj)ll+r % m+ 



where h 



,p , " P ,q^q I -w ^p, 



g r 



'PjQ 



dp><l) QptV>i+l • • • 



. (6) 



• 0) 

 ■ (8) 



To obtain (7) the multipliers applied to the rows of (4) 

 are zero for the first m rows excepting the />th; for the pth. 

 and the rows after the with they are the first minors of b p , q 

 corresponding to the members of its first column. The multi- 

 plier for ff m+r is minor corresponding to the r + l)th member 

 of the column, and is ( — l) r times the value with Op, m +i as 

 top row; while c p , m+r is ( — l) r-1 times the same value, that is 

 the multiplier is —c Ptm + r , the negative sign corresponding 

 to the fact that % p and f, n + r are on opposite sides of equa- 

 tion (7), on the same side in equations (4). 



§ 3. In a recent communication* it was shown that this 

 transformation of a homogeneous to a new heterogeneous 

 form, in which one group is represented by velocities, the 



* Phil. Mag. Julv 1908. 



