TRANSACTIONS OF SECTION A. 563 



then the flow takes place between surfaces represented by this equation. Thus 

 we can consider separately the flow in the two-dimensional sheet between consecu- 

 tive surfaces C and C + 80 ; and on the understanding that the equation of con- 

 tinuity is satisfied, the lines of this flow will be obtained by equating a stream func- 

 tion x//' to a constant. For, take any two points P and Q on the sheet ; the steady 

 flow per unit time across any curve PAQ on the sheet must be equal to that across 

 any other curve PBQ, provided there is no sink in the region between these curves 

 into which fluid can disappear. Thus the flow across any curve connecting P and 

 Q must be a function of the coordinates of these points, say F (P, Q) ; further, since 

 the flow across PQ is the sum of those across PB and BQ, this function must be of 

 the form V'(Q) - y}r(P). If therefore r denote at each point the thickness of the 

 sheet, and v the component velocity at right angles to the element of arc ds on it. 



j^Mvrdf« = ^/.(Q)-^/.(P;; 



that is MvrfZ.y, which is of the form G^d.Vj + G.^dx.^ + G^d.v.^ is the exact differen- 

 tial of a function \(r{a\, .Vn, a:^). Thus if one integral of a linear differential system 



d.v^ _ dx„ _ dx^ 



Ml Mj *<3 



is known the remaining one can be found by a quadrature whenever a value of 

 M is known which satisfies the equation of continuity 



dx^ dx^ dx^ 



This argument admits of immediate extension to hyper-space involving any 

 number n of Cartesian coordinates. In that case a knowledge of w — 2 integrals 

 determines an equal number of systems of hyper-surfaces along which the flow 

 takes place ; these divide the region into two-dimensional sheets, the flow in each 

 of which takes place independently and is determined by a stream function, as above, 

 whose general form can be determined by a quadrature. 



This is Jacobi's proposition. The conditions of its application are satisfied by 

 the special value M = 1 in the case of the differential equations of isoperimetrical 

 problems, including the general equations of dynamics. 



In all cases of course values of M exist, but it is only sometimes that they can 

 be analytically expressed. One method of trial is to express the determining 

 equation, after Jacobi, in the form 



8M ] /du^ . '^^2 . +'^""^ 



dx^ u\dx.^ dx., ' ' ' dxj' 



in which S represents a total differentiation. If then by means of the w — 2 known 

 integrals the quantity on the right-hand side can be expressed as a function of Aj 

 alone which is capable of integration, its integral is a form of M. 



In three dimensions the flow across a differential arc PQ, say ds on the sheet, 

 is equal to a lamellar element of volume whose projection on the plane 



x^x., is 



dx^ dxr. 



M., 



and whose height parallel to the axes of x^ is dx^, where 



dG=^dXr^; thus it is M-^r^-.iu./lx-^—u^dx^bG, which is accordingly an 

 exact differential in fZ.ij and rf.r^. In n dimensions it is similarly 



^idXr^dXj^ 



dx-y dx^ 



that is M(M./7.ri-Midr,)5C,8C, . . . 8C,.-o^'^^'^''^^ ' ' ' ^"-""^ -, 



d(x^x^ . . . x„) 



which is thus an exact differential when expressed in terms of .Cj and x.^ 



o o 



