242 NOTES. 



respectively. The rate at which the projection on x is increasing is 

 equal to the difference of the values of u&t the two ends, i.e. to 



c du ye du e du 



p dx p dy p dz 

 or, by (3), to 



Hence the projections of the line in question will at the time 

 t + dt be 



2x 



p p p 



respectively, that is, the line still forms part of a vortex-line. 

 Also, if ds be the length of this line at any instant, we have 



7 W 



as = e , 

 P 



where w is the angular velocity of the fluid. But if a- be the section of 

 a vortex-filament having ds as axis, the product pa-ds is constant with 

 regard to the time. Hence the strength coo- of the vortex is constant. 



This extension of Helmholtz proof, which is limited to the particular 

 case = 0, is due to Prof. Nanson. 



The connection between the above proof and that by Thomson given 

 in the text may be shewn as follows*. 



If A, B, C be the projections 011 the co-ordinate planes of the area of 

 a circuit moving with the fluid, we have 







where the integration is taken right round the circuit. If the circuit be 

 infinitely small, this gives 



dA /dv dw\ dv ~ dw 

 dt \dy dz) dx dx 



{Q A du dv n dw . . 



= AO - A - B - -- C -j- ..................... (4), 



dx dy dz 



where 6 has the same meaning as in the text. But (Art. 39) the 

 circulation in the circuit is equal to 2 (4 + -rjB + C), and since this is 

 constant with regard to the time we have 



........................ (5); 



Messenger of Mathematics, July, 1877. 



