A CERTAIN QUADRATIC FORM 239 



of zero area (straight line segments), points inside and outside this zero 

 conic describe their paths in opposite directions. 



The first case is that of the rotational homogeneous movement of the 

 points of a plane, the second is that of the harmonic homogeneous move- 

 ment. 



The object of introducing these particular displacement theorems is to 

 point out the solution of such a problem as, for example, the meteorological 

 one of the wind charts. At a series of points in a territory the direction 

 and velocity of the wind are observed. There are thus formed a network 

 of triangles such as P1P2P3. If we assume that the flow takes place under 

 the homogeneous law inside each triangle, then the direction and velocity 

 at each point is determined and interpolated. If we assume that the lines 

 of flow at the observed points are circles either direct or retrograde, which 

 will be more nearly the truth the smaller the triangles, then the lines of 

 flow for all particles can be mapped as continuous curves which are com- 

 posed of arcs of conies compounded at points where they cross the sides 

 of the triangles. 



On replacing A by d and ki by the m's, after dividing by df there results 



"S^^. T/.2 = MV2M ^ "^ 



nii 7^2 = MF2 -I- -i V iHi my Va^ (l9) 



an expression for the energy of a system of masses. The 2 term on the 

 right representing the mass-mean of the sum of the products of the rela- 

 tive momenta two and two. 



Differentiate (19) as to < and there results a corresponding relation in the 

 work of the forces along the paths. 



^Fasi=Fds+^^ nii mi ^ dst,-. (20) 



6. Let 



^d^x\^ , fd^y 



dt-/ ' \dt- ' 

 Similar to the equation in velocities 



72 = -zkiVi- - -LkihjVij^ (21) 



the equation in total accelerations 



o? = 'Zkiuc - '^kikj aii^ (22) 



results, and the corresponding force function when the m's are introduced. 



